Class 11 stars-bars MCQ Questions

Question 1/16 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(18) identical balls को (6) boxes में रखना है और exactly (3) boxes non-empty हों, तो count कौन-सी है?

(18) identical balls are placed into (6) boxes and exactly (3) boxes are non-empty. Which count is correct?

Explanation opens after your attempt

Correct Answer

A. \(^{6}C_3{}^{17}C_2\)

Step 1

Concept

First choose the (3) non-empty boxes and then split (18) balls into (3) positive parts. In exams use choose boxes plus positive stars and bars for exactly non-empty.

Step 2

Why this answer is correct

The correct answer is A. \(^{6}C_3{}^{17}C_2\). First choose the (3) non-empty boxes and then split (18) balls into (3) positive parts. In exams use choose boxes plus positive stars and bars for exactly non-empty.

Step 3

Exam Tip

पहले (3) non-empty boxes चुनें फिर (18) balls को (3) positive parts में बांटें। परीक्षा में exactly non-empty को choose boxes plus positive stars-bars करें।

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Expert
Level: 66

Question Hindi:
(18) identical balls को (6) boxes में रखना है और exactly (3) boxes non-empty हों, तो count कौन-सी है?

Question English:
(18) identical balls are placed into (6) boxes and exactly (3) boxes are non-empty. Which count is correct?

Options:
A. (^{6}C_3{}^{17}C_2)
B. (^{6}C_3{}^{20}C_2)
C. (^{23}C_5)
D. (6^{18})

Correct Answer:
A. (^{6}C_3{}^{17}C_2)

Explanation Hindi:
पहले (3) non-empty boxes चुनें फिर (18) balls को (3) positive parts में बांटें। परीक्षा में exactly non-empty को choose boxes plus positive stars-bars करें।

Explanation English:
First choose the (3) non-empty boxes and then split (18) balls into (3) positive parts. In exams use choose boxes plus positive stars and bars for exactly non-empty.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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- Formula/Math/Chemistry text must stay exactly readable; use LaTeX-style clean rendering where needed.

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Question 2/16 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(x_1+x_2+x_3+x_4=22\) में \(x_1\geq1\), \(x_2\geq2\), \(x_3\geq3\), \(x_4\geq4\) हो, तो count क्या है?

In \(x_1+x_2+x_3+x_4=22\), if \(x_1\geq1\), \(x_2\geq2\), \(x_3\geq3\), \(x_4\geq4\), what is the count?

Explanation opens after your attempt

Correct Answer

B. \(^{15}C_3\)

Step 1

Concept

After removing the minimum sum (10), (12) remains and is distributed among (4) variables. In exams subtract lower bounds and use stars and bars.

Step 2

Why this answer is correct

The correct answer is B. \(^{15}C_3\). After removing the minimum sum (10), (12) remains and is distributed among (4) variables. In exams subtract lower bounds and use stars and bars.

Step 3

Exam Tip

Minimum sum (10) हटाने पर (12) बचता है और (4) variables में distribute होता है। परीक्षा में lower bounds subtract करके stars and bars लगाएं।

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AI Video Prompt 16:9 + 9:16

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Expert
Level: 66

Question Hindi:
(x_1+x_2+x_3+x_4=22) में (x_1\geq1), (x_2\geq2), (x_3\geq3), (x_4\geq4) हो, तो count क्या है?

Question English:
In (x_1+x_2+x_3+x_4=22), if (x_1\geq1), (x_2\geq2), (x_3\geq3), (x_4\geq4), what is the count?

Options:
A. (^{25}C_3)
B. (^{15}C_3)
C. (^{22}C_4)
D. (4^{22})

Correct Answer:
B. (^{15}C_3)

Explanation Hindi:
Minimum sum (10) हटाने पर (12) बचता है और (4) variables में distribute होता है। परीक्षा में lower bounds subtract करके stars and bars लगाएं।

Explanation English:
After removing the minimum sum (10), (12) remains and is distributed among (4) variables. In exams subtract lower bounds and use stars and bars.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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- Use smooth transitions, no clutter, no watermark except Muft Shiksha branding.
- Formula/Math/Chemistry text must stay exactly readable; use LaTeX-style clean rendering where needed.

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Question 3/16 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(20) identical balls को (5) boxes में बांटना है और exactly (2) boxes empty हों। सही count कौन-सी है?

(20) identical balls are distributed into (5) boxes and exactly (2) boxes are empty. Which count is correct?

Explanation opens after your attempt

Correct Answer

A. \(^{5}C_2{}^{19}C_2\)

Step 1

Concept

First choose the empty boxes, then distribute positively into the remaining (3) boxes. In exams split exactly empty into box selection and positive distribution.

Step 2

Why this answer is correct

The correct answer is A. \(^{5}C_2{}^{19}C_2\). First choose the empty boxes, then distribute positively into the remaining (3) boxes. In exams split exactly empty into box selection and positive distribution.

Step 3

Exam Tip

पहले empty boxes चुनें, फिर बाकी (3) boxes में positive distribution करें। परीक्षा में exactly empty को boxes selection और positive distribution में तोड़ें।

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AI Video Prompt 16:9 + 9:16

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Question Page:
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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Expert
Level: 65

Question Hindi:
(20) identical balls को (5) boxes में बांटना है और exactly (2) boxes empty हों। सही count कौन-सी है?

Question English:
(20) identical balls are distributed into (5) boxes and exactly (2) boxes are empty. Which count is correct?

Options:
A. (^{5}C_2{}^{19}C_2)
B. (^{5}C_2{}^{22}C_2)
C. (5^{20})
D. (^{24}C_4)

Correct Answer:
A. (^{5}C_2{}^{19}C_2)

Explanation Hindi:
पहले empty boxes चुनें, फिर बाकी (3) boxes में positive distribution करें। परीक्षा में exactly empty को boxes selection और positive distribution में तोड़ें।

Explanation English:
First choose the empty boxes, then distribute positively into the remaining (3) boxes. In exams split exactly empty into box selection and positive distribution.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

Custom Support:
- Make 16:9 version for YouTube/Facebook.
- Make 9:16 version for Shorts/Reels.
- Add Hindi-English natural voice-over.
- Add readable subtitles.
- Add light background music, low volume.
- Keep all text inside safe margins.
- Use smooth transitions, no clutter, no watermark except Muft Shiksha branding.
- Formula/Math/Chemistry text must stay exactly readable; use LaTeX-style clean rendering where needed.

Output Needed:
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Question 4/16 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(x_1+x_2+x_3+x_4+x_5=12\) में exactly (3) variables positive हों, तो count क्या होगा?

In \(x_1+x_2+x_3+x_4+x_5=12\), if exactly (3) variables are positive, what is the count?

Explanation opens after your attempt

Correct Answer

A. \(^{5}C_3{}^{11}C_2\)

Step 1

Concept

Choose the (3) positive variables first, then split (12) into (3) positive parts. In exams use choose variables plus positive stars and bars for exactly positive variables.

Step 2

Why this answer is correct

The correct answer is A. \(^{5}C_3{}^{11}C_2\). Choose the (3) positive variables first, then split (12) into (3) positive parts. In exams use choose variables plus positive stars and bars for exactly positive variables.

Step 3

Exam Tip

पहले (3) positive variables चुनें, फिर (12) को (3) positive parts में बांटें। परीक्षा में exactly positive variables में choose variables plus positive stars-bars करें।

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AI Video Prompt 16:9 + 9:16

Is question ka premium MCQ video banane ke liye ready prompt. Copy karke Sora, Runway, Canva AI, CapCut AI, ChatGPT video workflow ya editor me use karein.

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Expert
Level: 65

Question Hindi:
(x_1+x_2+x_3+x_4+x_5=12) में exactly (3) variables positive हों, तो count क्या होगा?

Question English:
In (x_1+x_2+x_3+x_4+x_5=12), if exactly (3) variables are positive, what is the count?

Options:
A. (^{5}C_3{}^{11}C_2)
B. (^{5}C_3{}^{14}C_2)
C. (^{16}C_4)
D. (5^{12})

Correct Answer:
A. (^{5}C_3{}^{11}C_2)

Explanation Hindi:
पहले (3) positive variables चुनें, फिर (12) को (3) positive parts में बांटें। परीक्षा में exactly positive variables में choose variables plus positive stars-bars करें।

Explanation English:
Choose the (3) positive variables first, then split (12) into (3) positive parts. In exams use choose variables plus positive stars and bars for exactly positive variables.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

Custom Support:
- Make 16:9 version for YouTube/Facebook.
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- Add readable subtitles.
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- Keep all text inside safe margins.
- Use smooth transitions, no clutter, no watermark except Muft Shiksha branding.
- Formula/Math/Chemistry text must stay exactly readable; use LaTeX-style clean rendering where needed.

Output Needed:
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Question 5/16 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(x_1+x_2+x_3+x_4=30\) में \(x_1\geq2\), \(x_2\geq3\), \(x_3\geq4\), \(x_4\geq5\) हो, तो count क्या है?

In \(x_1+x_2+x_3+x_4=30\), if \(x_1\geq2\), \(x_2\geq3\), \(x_3\geq4\), \(x_4\geq5\), what is the count?

Explanation opens after your attempt

Correct Answer

A. \(^{19}C_3\)

Step 1

Concept

After removing the minimum sum (14), (16) remains, so \({}^{16+4-1}C_{3}\) is obtained. In exams subtract unequal lower bounds first.

Step 2

Why this answer is correct

The correct answer is A. \(^{19}C_3\). After removing the minimum sum (14), (16) remains, so \({}^{16+4-1}C_{3}\) is obtained. In exams subtract unequal lower bounds first.

Step 3

Exam Tip

Minimum sum (14) हटाने पर (16) बचता है, इसलिए \({}^{16+4-1}C_{3}\) मिलता है। परीक्षा में unequal lower bounds पहले subtract करें।

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AI Video Prompt 16:9 + 9:16

Is question ka premium MCQ video banane ke liye ready prompt. Copy karke Sora, Runway, Canva AI, CapCut AI, ChatGPT video workflow ya editor me use karein.

Create a premium AI MCQ quiz video for Muft Shiksha in BOTH formats: 16:9 landscape and 9:16 vertical short.

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Question Page:
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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Expert
Level: 65

Question Hindi:
(x_1+x_2+x_3+x_4=30) में (x_1\geq2), (x_2\geq3), (x_3\geq4), (x_4\geq5) हो, तो count क्या है?

Question English:
In (x_1+x_2+x_3+x_4=30), if (x_1\geq2), (x_2\geq3), (x_3\geq4), (x_4\geq5), what is the count?

Options:
A. (^{19}C_3)
B. (^{16}C_3)
C. (^{33}C_3)
D. (^{30}C_4)

Correct Answer:
A. (^{19}C_3)

Explanation Hindi:
Minimum sum (14) हटाने पर (16) बचता है, इसलिए ({}^{16+4-1}C_{3}) मिलता है। परीक्षा में unequal lower bounds पहले subtract करें।

Explanation English:
After removing the minimum sum (14), (16) remains, so ({}^{16+4-1}C_{3}) is obtained. In exams subtract unequal lower bounds first.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

Custom Support:
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Question 6/16 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(x_1+x_2+x_3+x_4=18\) में exactly (2) variables zero हों, तो count क्या है?

In \(x_1+x_2+x_3+x_4=18\), if exactly (2) variables are zero, what is the count?

Explanation opens after your attempt

Correct Answer

A. \(^{4}C_2{}^{17}C_1\)

Step 1

Concept

Choose the zero variables first, then the remaining two variables form a positive sum of (18). In exams use positive distribution for exactly-zero cases.

Step 2

Why this answer is correct

The correct answer is A. \(^{4}C_2{}^{17}C_1\). Choose the zero variables first, then the remaining two variables form a positive sum of (18). In exams use positive distribution for exactly-zero cases.

Step 3

Exam Tip

पहले zero variables चुनें, फिर बाकी दो variables positive sum (18) बनाते हैं। परीक्षा में exactly zero cases में positive distribution लगाएं।

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AI Video Prompt 16:9 + 9:16

Is question ka premium MCQ video banane ke liye ready prompt. Copy karke Sora, Runway, Canva AI, CapCut AI, ChatGPT video workflow ya editor me use karein.

Create a premium AI MCQ quiz video for Muft Shiksha in BOTH formats: 16:9 landscape and 9:16 vertical short.

Brand:
- Name: Muft Shiksha
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Question Page:
https://muftshiksha.com/question/in-x-1-x-2-x-3-x-4-18-if-exactly-2-variables-are-zero-what-is-the-count

MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Expert
Level: 64

Question Hindi:
(x_1+x_2+x_3+x_4=18) में exactly (2) variables zero हों, तो count क्या है?

Question English:
In (x_1+x_2+x_3+x_4=18), if exactly (2) variables are zero, what is the count?

Options:
A. (^{4}C_2{}^{17}C_1)
B. (^{4}C_2{}^{19}C_1)
C. (4^18)
D. (^{21}C_3)

Correct Answer:
A. (^{4}C_2{}^{17}C_1)

Explanation Hindi:
पहले zero variables चुनें, फिर बाकी दो variables positive sum (18) बनाते हैं। परीक्षा में exactly zero cases में positive distribution लगाएं।

Explanation English:
Choose the zero variables first, then the remaining two variables form a positive sum of (18). In exams use positive distribution for exactly-zero cases.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

Custom Support:
- Make 16:9 version for YouTube/Facebook.
- Make 9:16 version for Shorts/Reels.
- Add Hindi-English natural voice-over.
- Add readable subtitles.
- Add light background music, low volume.
- Keep all text inside safe margins.
- Use smooth transitions, no clutter, no watermark except Muft Shiksha branding.
- Formula/Math/Chemistry text must stay exactly readable; use LaTeX-style clean rendering where needed.

Output Needed:
- Final narration script.
- Scene-by-scene visual direction.
- Text overlays for 16:9 and 9:16.
- Thumbnail idea.
- Hashtags: #MuftShiksha #FreeEducation #MCQ #Quiz #Class11 #Study

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Question 7/16 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(x_1+x_2+x_3=20\) में \(x_1\geq3\), \(x_2\geq4\), \(x_3\geq5\) हो, तो count क्या है?

In \(x_1+x_2+x_3=20\), if \(x_1\geq3\), \(x_2\geq4\), \(x_3\geq5\), what is the count?

Explanation opens after your attempt

Correct Answer

A. \(^{10}C_2\)

Step 1

Concept

Removing the minimum (3+4+5=12) leaves (8). In exams shift unequal lower bounds first.

Step 2

Why this answer is correct

The correct answer is A. \(^{10}C_2\). Removing the minimum (3+4+5=12) leaves (8). In exams shift unequal lower bounds first.

Step 3

Exam Tip

Minimum (3+4+5=12) हटाने पर (8) बचता है। परीक्षा में unequal lower bounds को पहले shift करें।

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Expert
Level: 64

Question Hindi:
(x_1+x_2+x_3=20) में (x_1\geq3), (x_2\geq4), (x_3\geq5) हो, तो count क्या है?

Question English:
In (x_1+x_2+x_3=20), if (x_1\geq3), (x_2\geq4), (x_3\geq5), what is the count?

Options:
A. (^{10}C_2)
B. (^{8}C_2)
C. (^{12}C_2)
D. (^{22}C_2)

Correct Answer:
A. (^{10}C_2)

Explanation Hindi:
Minimum (3+4+5=12) हटाने पर (8) बचता है। परीक्षा में unequal lower bounds को पहले shift करें।

Explanation English:
Removing the minimum (3+4+5=12) leaves (8). In exams shift unequal lower bounds first.

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4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
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Question 8/16 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(x_1+x_2+x_3+x_4=25\) और \(x_i\geq2\) हो, तो solutions की संख्या कौन-सी है?

If \(x_1+x_2+x_3+x_4=25\) and \(x_i\geq2\), what is the number of solutions?

Explanation opens after your attempt

Correct Answer

B. \(^{20}C_3\)

Step 1

Concept

Give (2) first to the four variables, leaving (17). In exams subtract the lower bound and apply non-negative stars and bars.

Step 2

Why this answer is correct

The correct answer is B. \(^{20}C_3\). Give (2) first to the four variables, leaving (17). In exams subtract the lower bound and apply non-negative stars and bars.

Step 3

Exam Tip

चार variables को पहले (2) देने पर (17) बचता है। परीक्षा में lower bound घटाकर non-negative stars and bars लगाएं।

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Expert
Level: 64

Question Hindi:
(x_1+x_2+x_3+x_4=25) और (x_i\geq2) हो, तो solutions की संख्या कौन-सी है?

Question English:
If (x_1+x_2+x_3+x_4=25) and (x_i\geq2), what is the number of solutions?

Options:
A. (^{24}C_3)
B. (^{20}C_3)
C. (^{16}C_3)
D. (^{28}C_3)

Correct Answer:
B. (^{20}C_3)

Explanation Hindi:
चार variables को पहले (2) देने पर (17) बचता है। परीक्षा में lower bound घटाकर non-negative stars and bars लगाएं।

Explanation English:
Give (2) first to the four variables, leaving (17). In exams subtract the lower bound and apply non-negative stars and bars.

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Question 9/16 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(n) distinct objects से (r) objects repetition के साथ बिना order चुने जाएं, तो formula किससे आता है?

If (r) objects are chosen from (n) distinct objects with repetition and without order, which idea gives the formula?

Explanation opens after your attempt

Correct Answer

A. Stars and bars

Step 1

Concept

This is multiset selection and the answer is \({}^{n+r-1}C_r\). In exams handle repetition with no order by the bars method.

Step 2

Why this answer is correct

The correct answer is A. Stars and bars. This is multiset selection and the answer is \({}^{n+r-1}C_r\). In exams handle repetition with no order by the bars method.

Step 3

Exam Tip

यह multiset selection है और answer \({}^{n+r-1}C_r\) होता है। परीक्षा में repetition with no order को bars method से करें।

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Expert
Level: 64

Question Hindi:
(n) distinct objects से (r) objects repetition के साथ बिना order चुने जाएं, तो formula किससे आता है?

Question English:
If (r) objects are chosen from (n) distinct objects with repetition and without order, which idea gives the formula?

Options:
A. Stars and bars
B. Circular permutation
C. Derangement
D. Pascal triangle only

Correct Answer:
A. Stars and bars

Explanation Hindi:
यह multiset selection है और answer ({}^{n+r-1}C_r) होता है। परीक्षा में repetition with no order को bars method से करें।

Explanation English:
This is multiset selection and the answer is ({}^{n+r-1}C_r). In exams handle repetition with no order by the bars method.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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Question 10/16 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(6) identical prizes (10) students में बांटने हैं और एक student कई prizes पा सकता है। Count कौन-सी है?

(6) identical prizes are distributed among (10) students and one student may receive several prizes. Which count is correct?

Explanation opens after your attempt

Correct Answer

B. \(^{15}C_9\)

Step 1

Concept

In identical prize distribution, use (6) stars and (9) bars. In exams apply stars and bars for identical items and distinct receivers.

Step 2

Why this answer is correct

The correct answer is B. \(^{15}C_9\). In identical prize distribution, use (6) stars and (9) bars. In exams apply stars and bars for identical items and distinct receivers.

Step 3

Exam Tip

Identical prizes distribution में (6) stars और (9) bars लगते हैं। परीक्षा में identical items और distinct receivers में stars and bars लगाएं।

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AI Video Prompt 16:9 + 9:16

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Hard
Level: 66

Question Hindi:
(6) identical prizes (10) students में बांटने हैं और एक student कई prizes पा सकता है। Count कौन-सी है?

Question English:
(6) identical prizes are distributed among (10) students and one student may receive several prizes. Which count is correct?

Options:
A. (10^6)
B. (^{15}C_9)
C. (^{10}P_6)
D. (^{10}C_6)

Correct Answer:
B. (^{15}C_9)

Explanation Hindi:
Identical prizes distribution में (6) stars और (9) bars लगते हैं। परीक्षा में identical items और distinct receivers में stars and bars लगाएं।

Explanation English:
In identical prize distribution, use (6) stars and (9) bars. In exams apply stars and bars for identical items and distinct receivers.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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Question 11/16 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(14) identical coins को (4) persons में बांटना है और exactly (1) person को कुछ न मिले। Count क्या है?

(14) identical coins are distributed among (4) persons and exactly (1) person gets nothing. What is the count?

Explanation opens after your attempt

Correct Answer

A. \(^{4}C_1\cdot{}^{13}C_2\)

Step 1

Concept

First choose the empty person, then distribute positively among the remaining (3) persons. In exams use choose empty plus positive stars and bars for exactly empty conditions.

Step 2

Why this answer is correct

The correct answer is A. \(^{4}C_1\cdot{}^{13}C_2\). First choose the empty person, then distribute positively among the remaining (3) persons. In exams use choose empty plus positive stars and bars for exactly empty conditions.

Step 3

Exam Tip

पहले empty person चुनें, फिर बाकी (3) persons में positive distribution करें। परीक्षा में exactly empty condition में choose empty plus positive stars-bars लगाएं।

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AI Video Prompt 16:9 + 9:16

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Hard
Level: 66

Question Hindi:
(14) identical coins को (4) persons में बांटना है और exactly (1) person को कुछ न मिले। Count क्या है?

Question English:
(14) identical coins are distributed among (4) persons and exactly (1) person gets nothing. What is the count?

Options:
A. (^{4}C_1\cdot{}^{13}C_2)
B. (^{4}C_1\cdot{}^{16}C_2)
C. (4^{14})
D. (^{14}C_4)

Correct Answer:
A. (^{4}C_1\cdot{}^{13}C_2)

Explanation Hindi:
पहले empty person चुनें, फिर बाकी (3) persons में positive distribution करें। परीक्षा में exactly empty condition में choose empty plus positive stars-bars लगाएं।

Explanation English:
First choose the empty person, then distribute positively among the remaining (3) persons. In exams use choose empty plus positive stars and bars for exactly empty conditions.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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Question 12/16 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(11) identical balls को (5) boxes में रखना है और हर box में कम से कम (1) ball हो। Count कौन-सी है?

(11) identical balls are placed in (5) boxes and each box has at least (1) ball. Which count is correct?

Explanation opens after your attempt

Correct Answer

A. \(^{10}C_4\)

Step 1

Concept

After giving (1) ball to each box, (6) balls remain. In exams allot the minimum first in positive distribution.

Step 2

Why this answer is correct

The correct answer is A. \(^{10}C_4\). After giving (1) ball to each box, (6) balls remain. In exams allot the minimum first in positive distribution.

Step 3

Exam Tip

हर box को (1) ball देने के बाद (6) balls बचती हैं। परीक्षा में positive distribution में पहले minimum allot करें।

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Hard
Level: 66

Question Hindi:
(11) identical balls को (5) boxes में रखना है और हर box में कम से कम (1) ball हो। Count कौन-सी है?

Question English:
(11) identical balls are placed in (5) boxes and each box has at least (1) ball. Which count is correct?

Options:
A. (^{10}C_4)
B. (^{15}C_4)
C. (5^{11})
D. (^{11}C_5)

Correct Answer:
A. (^{10}C_4)

Explanation Hindi:
हर box को (1) ball देने के बाद (6) balls बचती हैं। परीक्षा में positive distribution में पहले minimum allot करें।

Explanation English:
After giving (1) ball to each box, (6) balls remain. In exams allot the minimum first in positive distribution.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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- Use smooth transitions, no clutter, no watermark except Muft Shiksha branding.
- Formula/Math/Chemistry text must stay exactly readable; use LaTeX-style clean rendering where needed.

Output Needed:
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Question 13/16 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(9) identical pens को (4) students में बांटना है और खाली हिस्सा allowed है। Count कौन-सी है?

(9) identical pens are to be distributed among (4) students and zero share is allowed. Which count is correct?

Explanation opens after your attempt

Correct Answer

B. \(^{12}C_3\)

Step 1

Concept

For identical pens and distinct students, use (9) stars and (3) bars. In exams remember stars and bars for zero-allowed distribution.

Step 2

Why this answer is correct

The correct answer is B. \(^{12}C_3\). For identical pens and distinct students, use (9) stars and (3) bars. In exams remember stars and bars for zero-allowed distribution.

Step 3

Exam Tip

Identical pens और distinct students के लिए (9) stars तथा (3) bars लगते हैं। परीक्षा में zero allowed distribution में stars and bars याद रखें।

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AI Video Prompt 16:9 + 9:16

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Hard
Level: 66

Question Hindi:
(9) identical pens को (4) students में बांटना है और खाली हिस्सा allowed है। Count कौन-सी है?

Question English:
(9) identical pens are to be distributed among (4) students and zero share is allowed. Which count is correct?

Options:
A. (4^9)
B. (^{12}C_3)
C. (^{9}C_4)
D. (^{9}P_4)

Correct Answer:
B. (^{12}C_3)

Explanation Hindi:
Identical pens और distinct students के लिए (9) stars तथा (3) bars लगते हैं। परीक्षा में zero allowed distribution में stars and bars याद रखें।

Explanation English:
For identical pens and distinct students, use (9) stars and (3) bars. In exams remember stars and bars for zero-allowed distribution.

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4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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Question 14/16 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(x_1+x_2+x_3=21\) और \(x_i\geq3\) हो, तो हलों की संख्या क्या होगी?

If \(x_1+x_2+x_3=21\) and \(x_i\geq3\), what is the number of solutions?

Explanation opens after your attempt

Correct Answer

A. \(^{14}C_2\)

Step 1

Concept

Giving (3) first to each variable leaves (12), then count non-negative solutions. In exams shift the minimum condition.

Step 2

Why this answer is correct

The correct answer is A. \(^{14}C_2\). Giving (3) first to each variable leaves (12), then count non-negative solutions. In exams shift the minimum condition.

Step 3

Exam Tip

हर variable को पहले (3) देने पर (12) बचता है, फिर अऋणात्मक हल गिनते हैं। परीक्षा में minimum condition को shift करें।

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Hard
Level: 66

Question Hindi:
(x_1+x_2+x_3=21) और (x_i\geq3) हो, तो हलों की संख्या क्या होगी?

Question English:
If (x_1+x_2+x_3=21) and (x_i\geq3), what is the number of solutions?

Options:
A. (^{14}C_2)
B. (^{20}C_2)
C. (^{23}C_2)
D. (^{18}C_3)

Correct Answer:
A. (^{14}C_2)

Explanation Hindi:
हर variable को पहले (3) देने पर (12) बचता है, फिर अऋणात्मक हल गिनते हैं। परीक्षा में minimum condition को shift करें।

Explanation English:
Giving (3) first to each variable leaves (12), then count non-negative solutions. In exams shift the minimum condition.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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Question 15/16 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(x_1+x_2+x_3+x_4=18\) के अऋणात्मक पूर्णांक हलों की संख्या क्या है?

What is the number of non-negative integer solutions of \(x_1+x_2+x_3+x_4=18\)?

Explanation opens after your attempt

Correct Answer

B. \(^{21}C_3\)

Step 1

Concept

In stars and bars, (18) stars and (3) bars are arranged. In exams use \(^{n+r-1}C_{r-1}\) for non-negative solutions.

Step 2

Why this answer is correct

The correct answer is B. \(^{21}C_3\). In stars and bars, (18) stars and (3) bars are arranged. In exams use \(^{n+r-1}C_{r-1}\) for non-negative solutions.

Step 3

Exam Tip

Stars and bars में (18) stars और (3) bars arrange होते हैं। परीक्षा में अऋणात्मक हलों के लिए \(^{n+r-1}C_{r-1}\) लगाएं।

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Derivations of formulas and their connections
Difficulty: Hard
Level: 66

Question Hindi:
(x_1+x_2+x_3+x_4=18) के अऋणात्मक पूर्णांक हलों की संख्या क्या है?

Question English:
What is the number of non-negative integer solutions of (x_1+x_2+x_3+x_4=18)?

Options:
A. (^{18}C_4)
B. (^{21}C_3)
C. (^{17}C_3)
D. (4^{18})

Correct Answer:
B. (^{21}C_3)

Explanation Hindi:
Stars and bars में (18) stars और (3) bars arrange होते हैं। परीक्षा में अऋणात्मक हलों के लिए (^{n+r-1}C_{r-1}) लगाएं।

Explanation English:
In stars and bars, (18) stars and (3) bars are arranged. In exams use (^{n+r-1}C_{r-1}) for non-negative solutions.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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Question 16/16 Expert Mathematics Permutations and Combinations Fundamental principle of counting Class 11 Level 52

(10) समान सिक्के (4) बच्चों में बाँटने हैं। हर बच्चे को कम से कम (1) सिक्का मिले और सबसे बड़े बच्चे को अधिकतम (4) सिक्के मिलें। कुल कितने तरीके होंगे?

(10) identical coins are to be distributed among (4) children. Each child must get at least (1) coin, and the eldest child can get at most (4) coins. How many distributions are possible?

Explanation opens after your attempt

Correct Answer

C. (74)

Step 1

Concept

First count positive solutions and subtract solutions where the eldest child gets (5) or more coins. Complement counting is useful for upper-bound restrictions.

Step 2

Why this answer is correct

The correct answer is C. (74). First count positive solutions and subtract solutions where the eldest child gets (5) or more coins. Complement counting is useful for upper-bound restrictions.

Step 3

Exam Tip

पहले धनात्मक हलों की संख्या गिनें और फिर बड़े बच्चे को (5) या अधिक सिक्के मिलने वाले हल घटाएँ। सीमा वाली शर्त में पूरक विधि उपयोगी है।

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MCQ Data:
Class: Class 11
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Fundamental principle of counting
Difficulty: Expert
Level: 52

Question Hindi:
(10) समान सिक्के (4) बच्चों में बाँटने हैं। हर बच्चे को कम से कम (1) सिक्का मिले और सबसे बड़े बच्चे को अधिकतम (4) सिक्के मिलें। कुल कितने तरीके होंगे?

Question English:
(10) identical coins are to be distributed among (4) children. Each child must get at least (1) coin, and the eldest child can get at most (4) coins. How many distributions are possible?

Options:
A. (64)
B. (70)
C. (74)
D. (80)

Correct Answer:
C. (74)

Explanation Hindi:
पहले धनात्मक हलों की संख्या गिनें और फिर बड़े बच्चे को (5) या अधिक सिक्के मिलने वाले हल घटाएँ। सीमा वाली शर्त में पूरक विधि उपयोगी है।

Explanation English:
First count positive solutions and subtract solutions where the eldest child gets (5) or more coins. Complement counting is useful for upper-bound restrictions.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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stars-bars MCQ Questions for Class 11

Practice Questions

(18) identical balls को (6) boxes में रखना है और exactly (3) boxes non-empty हों, तो count कौन-सी है?

Concept

Why this answer is correct

Exam Tip

\(x_1+x_2+x_3+x_4=22\) में \(x_1\geq1\), \(x_2\geq2\), \(x_3\geq3\), \(x_4\geq4\) हो, तो count क्या है?

Concept

Why this answer is correct

Exam Tip

(20) identical balls को (5) boxes में बांटना है और exactly (2) boxes empty हों। सही count कौन-सी है?

Concept

Why this answer is correct

Exam Tip

\(x_1+x_2+x_3+x_4+x_5=12\) में exactly (3) variables positive हों, तो count क्या होगा?

Concept

Why this answer is correct

Exam Tip

\(x_1+x_2+x_3+x_4=30\) में \(x_1\geq2\), \(x_2\geq3\), \(x_3\geq4\), \(x_4\geq5\) हो, तो count क्या है?

Concept

Why this answer is correct

Exam Tip

\(x_1+x_2+x_3+x_4=18\) में exactly (2) variables zero हों, तो count क्या है?

Concept

Why this answer is correct

Exam Tip

\(x_1+x_2+x_3=20\) में \(x_1\geq3\), \(x_2\geq4\), \(x_3\geq5\) हो, तो count क्या है?

Concept

Why this answer is correct

Exam Tip

\(x_1+x_2+x_3+x_4=25\) और \(x_i\geq2\) हो, तो solutions की संख्या कौन-सी है?

Concept

Why this answer is correct

Exam Tip

(n) distinct objects से (r) objects repetition के साथ बिना order चुने जाएं, तो formula किससे आता है?

Concept

Why this answer is correct

Exam Tip

(6) identical prizes (10) students में बांटने हैं और एक student कई prizes पा सकता है। Count कौन-सी है?

Concept

Why this answer is correct

Exam Tip

(14) identical coins को (4) persons में बांटना है और exactly (1) person को कुछ न मिले। Count क्या है?

Concept

Why this answer is correct

Exam Tip

(11) identical balls को (5) boxes में रखना है और हर box में कम से कम (1) ball हो। Count कौन-सी है?

Concept

Why this answer is correct

Exam Tip

(9) identical pens को (4) students में बांटना है और खाली हिस्सा allowed है। Count कौन-सी है?

Concept

Why this answer is correct

Exam Tip

\(x_1+x_2+x_3=21\) और \(x_i\geq3\) हो, तो हलों की संख्या क्या होगी?

Concept

Why this answer is correct

Exam Tip

\(x_1+x_2+x_3+x_4=18\) के अऋणात्मक पूर्णांक हलों की संख्या क्या है?

Concept

Why this answer is correct

Exam Tip

Concept

Why this answer is correct

Exam Tip

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