एक पहचान कोड में (3) अक्षर और (1) अंक है। अक्षरों के लिए (5) विकल्प और अंक के लिए (8) विकल्प हैं। पुनरावृत्ति सहित कितने कोड बनेंगे?
An identity code has (3) letters and (1) digit. There are (5) choices for each letter and (8) choices for the digit. With repetition allowed, how many codes are possible?
#permutations and combinations
#counting principle
#codes
A (40)
B (125)
C (625)
D (1000)
Explanation opens after your attempt
Step 1
Concept
Total choices are \(5\times5\times5\times8=1000\). In mixed codes, write choices for each position separately.
Step 2
Why this answer is correct
The correct answer is D. (1000). Total choices are \(5\times5\times5\times8=1000\). In mixed codes, write choices for each position separately.
Step 3
Exam Tip
कुल विकल्प \(5\times5\times5\times8=1000\) हैं। मिश्रित कोड में हर स्थान का विकल्प अलग लिखें।
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एक सिक्का उछालने और एक पासा फेंकने पर कुल परिणाम कितने होंगे?
How many total outcomes are possible when a coin is tossed and a die is rolled?
#permutations and combinations
#counting principle
#outcomes
A (12)
B (8)
C (6)
D (2)
Explanation opens after your attempt
Step 1
Concept
A coin has (2) outcomes and a die has (6), so \(2\times6=12\). Multiply when two experiments are performed together.
Step 2
Why this answer is correct
The correct answer is A. (12). A coin has (2) outcomes and a die has (6), so \(2\times6=12\). Multiply when two experiments are performed together.
Step 3
Exam Tip
सिक्के के (2) और पासे के (6) परिणाम हैं इसलिए \(2\times6=12\)। दो प्रयोग साथ हों तो गुणन करें।
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दो अलग-अलग सिक्के उछाले जाते हैं। कुल परिणाम कितने होंगे?
Two distinct coins are tossed. How many total outcomes are possible?
#permutations and combinations
#counting principle
#outcomes
A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
Each coin has (2) outcomes, so \(2\times2=4\). Treat distinct coins as separate stages.
Step 2
Why this answer is correct
The correct answer is B. (4). Each coin has (2) outcomes, so \(2\times2=4\). Treat distinct coins as separate stages.
Step 3
Exam Tip
हर सिक्के के (2) परिणाम हैं इसलिए \(2\times2=4\)। अलग-अलग सिक्कों को अलग चरण मानें।
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दो पासे एक साथ फेंके जाते हैं। कुल क्रमित परिणाम कितने होंगे?
Two dice are rolled together. How many ordered outcomes are possible?
#permutations and combinations
#counting principle
#outcomes
A (12)
B (18)
C (36)
D (72)
Explanation opens after your attempt
Step 1
Concept
The first die has (6) outcomes and the second has (6), so \(6\times6=36\). In ordered outcomes, the roles of both dice are distinct.
Step 2
Why this answer is correct
The correct answer is C. (36). The first die has (6) outcomes and the second has (6), so \(6\times6=36\). In ordered outcomes, the roles of both dice are distinct.
Step 3
Exam Tip
पहले पासे के (6) और दूसरे के (6) परिणाम हैं इसलिए \(6\times6=36\)। क्रमित परिणाम में दोनों पासों की भूमिका अलग रहती है।
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एक चक्र में (4) रंग और (3) आकार उपलब्ध हैं। एक रंग और एक आकार चुनकर डिज़ाइन कितने तरीकों से बनेगा?
A wheel has (4) colors and (3) shapes available. In how many ways can a design be made by choosing one color and one shape?
#permutations and combinations
#counting principle
#multiplication rule
A (7)
B (9)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
One color and one shape are needed, so \(4\times3=12\). Multiply two independent attributes.
Step 2
Why this answer is correct
The correct answer is D. (12). One color and one shape are needed, so \(4\times3=12\). Multiply two independent attributes.
Step 3
Exam Tip
एक रंग और एक आकार चाहिए इसलिए \(4\times3=12\)। दो स्वतंत्र विशेषताओं को गुणा करें।
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एक पासवर्ड में पहले (3) अक्षरों में से एक और फिर (4) अंकों में से एक चुनना है। कितने पासवर्ड बनेंगे?
A password is made by choosing one of (3) letters first and then one of (4) digits. How many passwords are possible?
#permutations and combinations
#counting principle
#password
A (12)
B (7)
C (9)
D (16)
Explanation opens after your attempt
Step 1
Concept
The first stage has (3) choices and the second has (4), so \(3\times4=12\). Fill password positions in order.
Step 2
Why this answer is correct
The correct answer is A. (12). The first stage has (3) choices and the second has (4), so \(3\times4=12\). Fill password positions in order.
Step 3
Exam Tip
पहले चरण में (3) और दूसरे में (4) विकल्प हैं इसलिए \(3\times4=12\)। पासवर्ड के स्थानों को क्रम से भरें।
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एक प्रतियोगिता में (5) कविता विषय और (6) चित्रकला विषय हैं। विद्यार्थी केवल एक विषय चुन सकता है। चयन के कितने तरीके हैं?
In a competition, there are (5) poetry topics and (6) painting topics. A student can choose only one topic. How many choices are possible?
#permutations and combinations
#counting principle
#addition rule
A (30)
B (11)
C (6)
D (5)
Explanation opens after your attempt
Step 1
Concept
Only one topic is to be chosen, so (5+6=11). Add when option groups are separate and not chosen together.
Step 2
Why this answer is correct
The correct answer is B. (11). Only one topic is to be chosen, so (5+6=11). Add when option groups are separate and not chosen together.
Step 3
Exam Tip
केवल एक विषय चुनना है इसलिए (5+6=11)। विकल्प समूह अलग हों और साथ नहीं चुने जाते हों तो जोड़ें।
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एक स्कूल यात्रा के लिए (3) संग्रहालय या (4) पार्क में से एक स्थान चुनना है। कुल विकल्प कितने हैं?
For a school trip, one place is to be chosen from (3) museums or (4) parks. How many total options are there?
#permutations and combinations
#counting principle
#addition rule
A (12)
B (9)
C (7)
D (1)
Explanation opens after your attempt
Step 1
Concept
The place will be a museum or a park, so (3+4=7). Use the addition rule for separate “or” choices.
Step 2
Why this answer is correct
The correct answer is C. (7). The place will be a museum or a park, so (3+4=7). Use the addition rule for separate “or” choices.
Step 3
Exam Tip
स्थान संग्रहालय या पार्क में से होगा इसलिए (3+4=7)। “या” वाले अलग चयन में योग का नियम लगाएँ।
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एक ऐप में (4) भाषा विकल्प और (2) थीम विकल्प हैं। एक भाषा और एक थीम चुनने के कितने तरीके हैं?
An app has (4) language options and (2) theme options. In how many ways can one language and one theme be selected?
#permutations and combinations
#counting principle
#multiplication rule
A (6)
B (4)
C (2)
D (8)
Explanation opens after your attempt
Step 1
Concept
Both selections are made together, so \(4\times2=8\). Multiply independent setting options.
Step 2
Why this answer is correct
The correct answer is D. (8). Both selections are made together, so \(4\times2=8\). Multiply independent setting options.
Step 3
Exam Tip
दोनों चयन साथ करने हैं इसलिए \(4\times2=8\)। सेटिंग्स के स्वतंत्र विकल्पों को गुणा करें।
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एक छात्र (2) बैग, (5) पानी की बोतल और (3) टिफिन में से एक-एक चीज़ चुनता है। कुल तरीके कितने होंगे?
A student chooses one item each from (2) bags, (5) water bottles, and (3) lunch boxes. How many total ways are possible?
#permutations and combinations
#counting principle
#multiplication rule
A (30)
B (10)
C (15)
D (25)
Explanation opens after your attempt
Step 1
Concept
Total ways are \(2\times5\times3=30\). Multiply when one item is chosen from each category.
Step 2
Why this answer is correct
The correct answer is A. (30). Total ways are \(2\times5\times3=30\). Multiply when one item is chosen from each category.
Step 3
Exam Tip
कुल तरीके \(2\times5\times3=30\) हैं। एक-एक चीज़ अलग श्रेणियों से हो तो गुणा करें।
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एक परीक्षा कोड में (2) अंकों के बाद (1) अक्षर है। अंक (0) से (9) तक और अक्षर (5) दिए गए हैं। पुनरावृत्ति सहित कितने कोड बनेंगे?
An exam code has (2) digits followed by (1) letter. Digits are from (0) to (9) and (5) letters are given. With repetition allowed, how many codes can be formed?
#permutations and combinations
#counting principle
#codes
A (50)
B (500)
C (100)
D (250)
Explanation opens after your attempt
Step 1
Concept
The two digit positions have (10) and (10) choices, and the letter has (5) choices. Total ways are \(10\times10\times5=500\).
Step 2
Why this answer is correct
The correct answer is B. (500). The two digit positions have (10) and (10) choices, and the letter has (5) choices. Total ways are \(10\times10\times5=500\).
Step 3
Exam Tip
दो अंक स्थानों के (10) और (10) विकल्प तथा अक्षर के (5) विकल्प हैं। कुल \(10\times10\times5=500\) हैं।
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एक (4)-अंकीय पिन में प्रत्येक स्थान पर (1) से (6) तक अंक आ सकते हैं। पुनरावृत्ति की अनुमति है तो कुल पिन कितने हैं?
In a (4)-digit PIN, each position can have digits from (1) to (6). If repetition is allowed, how many PINs are possible?
#permutations and combinations
#counting principle
#pin
A (24)
B (1296)
C (144)
D (720)
Explanation opens after your attempt
Step 1
Concept
Each position has (6) choices, so \(6^4=1296\). When the same choices repeat, using powers is faster.
Step 2
Why this answer is correct
The correct answer is B. (1296). Each position has (6) choices, so \(6^4=1296\). When the same choices repeat, using powers is faster.
Step 3
Exam Tip
प्रत्येक स्थान के (6) विकल्प हैं इसलिए \(6^4=1296\)। समान विकल्प बार-बार हों तो घात का प्रयोग तेज होता है।
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अंकों (1,2,3,4) से पुनरावृत्ति बिना (4)-अंकीय संख्याएँ कितनी बनेंगी?
How many (4)-digit numbers can be formed from digits (1,2,3,4) without repetition?
#permutations and combinations
#counting principle
#digits
A (16)
B (12)
C (24)
D (64)
Explanation opens after your attempt
Step 1
Concept
The four positions have (4,3,2,1) choices, so \(4\times3\times2\times1=24\). If all digits are used, choices decrease step by step.
Step 2
Why this answer is correct
The correct answer is C. (24). The four positions have (4,3,2,1) choices, so \(4\times3\times2\times1=24\). If all digits are used, choices decrease step by step.
Step 3
Exam Tip
चार स्थानों के विकल्प (4,3,2,1) हैं इसलिए \(4\times3\times2\times1=24\)। सभी अंक उपयोग हों तो विकल्प क्रम से घटते हैं।
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अंकों (0,1,2,3) से पुनरावृत्ति बिना (3)-अंकीय सम संख्याएँ कितनी बनेंगी?
How many (3)-digit even numbers can be formed from digits (0,1,2,3) without repetition?
#permutations and combinations
#counting principle
#even numbers
A (4)
B (8)
C (10)
D (10)
Explanation opens after your attempt
Step 1
Concept
If the units digit is (0), there are \(3\times2=6\) ways, and if it is (2), there are \(2\times2=4\) ways. Total ways are (6+4=10).
Step 2
Why this answer is correct
The correct answer is B. (8). If the units digit is (0), there are \(3\times2=6\) ways, and if it is (2), there are \(2\times2=4\) ways. Total ways are (6+4=10).
Step 3
Exam Tip
यदि इकाई (0) है तो \(3\times2=6\) तरीके हैं और यदि इकाई (2) है तो सैकड़े के (2) तथा दहाई के (2) तरीके हैं। कुल (6+4=10) हैं।
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एक कैफे में (3) प्रकार की चाय, (4) प्रकार की कॉफी और (2) प्रकार का जूस है। केवल एक पेय चुनने के कितने तरीके हैं?
A cafe has (3) types of tea, (4) types of coffee, and (2) types of juice. In how many ways can only one drink be chosen?
#permutations and combinations
#counting principle
#addition rule
A (24)
B (12)
C (9)
D (7)
Explanation opens after your attempt
Step 1
Concept
Only one drink is chosen, so (3+4+2=9). Add when the groups are alternatives, not simultaneous choices.
Step 2
Why this answer is correct
The correct answer is C. (9). Only one drink is chosen, so (3+4+2=9). Add when the groups are alternatives, not simultaneous choices.
Step 3
Exam Tip
केवल एक पेय चुनना है इसलिए (3+4+2=9)। एक ही श्रेणी नहीं बल्कि वैकल्पिक समूह हों तो जोड़ें।
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एक टीम जर्सी के लिए (5) रंग और (4) आकार उपलब्ध हैं। एक रंग और एक आकार चुनने के कितने तरीके हैं?
For a team jersey, (5) colors and (4) sizes are available. In how many ways can one color and one size be chosen?
#permutations and combinations
#counting principle
#multiplication rule
A (9)
B (16)
C (25)
D (20)
Explanation opens after your attempt
Step 1
Concept
Both color and size are to be selected, so \(5\times4=20\). Multiply when product features are independent.
Step 2
Why this answer is correct
The correct answer is D. (20). Both color and size are to be selected, so \(5\times4=20\). Multiply when product features are independent.
Step 3
Exam Tip
रंग और आकार दोनों चुनने हैं इसलिए \(5\times4=20\)। उत्पाद की अलग विशेषताएँ स्वतंत्र हों तो गुणा करें।
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एक बैज में (3) पृष्ठभूमि रंग, (2) प्रतीक और (4) बॉर्डर विकल्प हैं। एक बैज कितने तरीकों से बनेगा?
A badge has (3) background colors, (2) symbols, and (4) border options. In how many ways can one badge be made?
#permutations and combinations
#counting principle
#multiplication rule
A (24)
B (9)
C (18)
D (12)
Explanation opens after your attempt
Step 1
Concept
The three stages are independent, so \(3\times2\times4=24\). In design questions, treat each feature as one stage.
Step 2
Why this answer is correct
The correct answer is A. (24). The three stages are independent, so \(3\times2\times4=24\). In design questions, treat each feature as one stage.
Step 3
Exam Tip
तीनों चरण स्वतंत्र हैं इसलिए \(3\times2\times4=24\)। डिज़ाइन प्रश्नों में हर फीचर को एक चरण मानें।
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एक छात्र (6) ऑनलाइन कोर्स या (3) ऑफलाइन कोर्स में से केवल एक कोर्स चुनता है। कुल चयन कितने हैं?
A student chooses only one course from (6) online courses or (3) offline courses. How many total choices are there?
#permutations and combinations
#counting principle
#addition rule
A (18)
B (9)
C (6)
D (3)
Explanation opens after your attempt
Step 1
Concept
Only one course is to be chosen, so (6+3=9). Add when the options are not taken together.
Step 2
Why this answer is correct
The correct answer is B. (9). Only one course is to be chosen, so (6+3=9). Add when the options are not taken together.
Step 3
Exam Tip
एक ही कोर्स चुनना है इसलिए (6+3=9)। विकल्पों को एक साथ नहीं लेना हो तो योग करें।
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एक कार्ड पर (4) चित्र विकल्प और (5) संदेश विकल्प हैं। एक चित्र और एक संदेश चुनकर कार्ड कितने तरीकों से बनेगा?
A card has (4) picture options and (5) message options. In how many ways can a card be made by choosing one picture and one message?
#permutations and combinations
#counting principle
#multiplication rule
A (9)
B (16)
C (20)
D (25)
Explanation opens after your attempt
Step 1
Concept
One picture and one message are both needed, so \(4\times5=20\). The multiplication principle applies to combined selection.
Step 2
Why this answer is correct
The correct answer is C. (20). One picture and one message are both needed, so \(4\times5=20\). The multiplication principle applies to combined selection.
Step 3
Exam Tip
एक चित्र और एक संदेश दोनों चाहिए इसलिए \(4\times5=20\)। संयुक्त चयन में गुणा सिद्धांत लागू होता है।
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एक पेन ड्राइव का रंग (3) विकल्पों में से और क्षमता (4) विकल्पों में से चुनी जाती है। कुल मॉडल कितने बनेंगे?
A pen drive color is chosen from (3) options and capacity from (4) options. How many total models can be formed?
#permutations and combinations
#counting principle
#multiplication rule
A (7)
B (9)
C (16)
D (12)
Explanation opens after your attempt
Step 1
Concept
Total models are \(3\times4=12\). In product models, multiply each independent feature.
Step 2
Why this answer is correct
The correct answer is D. (12). Total models are \(3\times4=12\). In product models, multiply each independent feature.
Step 3
Exam Tip
कुल मॉडल \(3\times4=12\) होंगे। उत्पाद मॉडल में हर स्वतंत्र विशेषता को गुणा करें।
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एक कक्षा में (4) मॉनिटर उम्मीदवार और (3) सह-मॉनिटर उम्मीदवार हैं। एक मॉनिटर और एक सह-मॉनिटर चुनने के कितने तरीके हैं?
In a class, there are (4) monitor candidates and (3) assistant monitor candidates. In how many ways can one monitor and one assistant monitor be chosen?
#permutations and combinations
#counting principle
#selection
A (12)
B (7)
C (10)
D (24)
Explanation opens after your attempt
Step 1
Concept
Two different posts are to be filled, so \(4\times3=12\). Treat selections for different posts as ordered stages.
Step 2
Why this answer is correct
The correct answer is A. (12). Two different posts are to be filled, so \(4\times3=12\). Treat selections for different posts as ordered stages.
Step 3
Exam Tip
दो अलग पद भरने हैं इसलिए \(4\times3=12\)। अलग पदों के चयन को क्रमबद्ध चरण मानें।
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एक व्यक्ति (5) बसों में से एक से स्टेशन जाता है और फिर (2) ऑटो में से एक से घर जाता है। कुल यात्रा तरीके कितने हैं?
A person takes one of (5) buses to the station and then one of (2) autos to home. How many total travel ways are possible?
#permutations and combinations
#counting principle
#routes
A (7)
B (10)
C (5)
D (2)
Explanation opens after your attempt
Step 1
Concept
There are (5) choices first and (2) choices next, so \(5\times2=10\). Multiply the stages in a sequential journey.
Step 2
Why this answer is correct
The correct answer is B. (10). There are (5) choices first and (2) choices next, so \(5\times2=10\). Multiply the stages in a sequential journey.
Step 3
Exam Tip
पहले (5) और फिर (2) विकल्प हैं इसलिए \(5\times2=10\)। क्रम से होने वाली यात्रा में चरणों को गुणा करें।
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एक वेबसाइट साइनअप में (3) प्रोफाइल फोटो विकल्प और (6) यूज़रनेम स्टाइल विकल्प हैं। एक प्रोफाइल बनाने के कितने तरीके हैं?
A website signup has (3) profile photo options and (6) username style options. In how many ways can one profile be made?
#permutations and combinations
#counting principle
#multiplication rule
A (9)
B (12)
C (18)
D (36)
Explanation opens after your attempt
Step 1
Concept
Both choices are independent, so \(3\times6=18\). In profile setup, multiply every required selection.
Step 2
Why this answer is correct
The correct answer is C. (18). Both choices are independent, so \(3\times6=18\). In profile setup, multiply every required selection.
Step 3
Exam Tip
दोनों चुनाव स्वतंत्र हैं इसलिए \(3\times6=18\)। प्रोफाइल सेटअप में हर अनिवार्य चयन को गुणा करें।
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एक गेम अवतार में (2) चेहरे, (4) टोपी और (3) पोशाक विकल्प हैं। एक अवतार कितने तरीकों से बनाया जा सकता है?
A game avatar has (2) face options, (4) hat options, and (3) costume options. In how many ways can one avatar be created?
#permutations and combinations
#counting principle
#multiplication rule
A (9)
B (12)
C (18)
D (24)
Explanation opens after your attempt
Step 1
Concept
Total avatars are \(2\times4\times3=24\). When there are three independent features, multiply all choices.
Step 2
Why this answer is correct
The correct answer is D. (24). Total avatars are \(2\times4\times3=24\). When there are three independent features, multiply all choices.
Step 3
Exam Tip
कुल अवतार \(2\times4\times3=24\) होंगे। तीन स्वतंत्र फीचर हों तो सभी विकल्पों को गुणा करें।
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एक छात्र के पास (4) कमीज़ और (3) पैंट हैं। वह एक कमीज़ और एक पैंट कितने तरीकों से चुन सकता है?
A student has (4) shirts and (3) trousers. In how many ways can he choose one shirt and one trouser?
#permutations and combinations
#counting principle
#multiplication rule
A (7)
B (9)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
By the multiplication principle total ways are \(4\times3=12\). In exams, “and” usually indicates multiplication.
Step 2
Why this answer is correct
The correct answer is C. (12). By the multiplication principle total ways are \(4\times3=12\). In exams, “and” usually indicates multiplication.
Step 3
Exam Tip
गुणन सिद्धांत से कुल तरीके \(4\times3=12\) हैं। परीक्षा में “और” आने पर सामान्यतः गुणा करें।
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एक मेन्यू में (5) नाश्ते और (4) पेय हैं। एक नाश्ता और एक पेय चुनने के कितने तरीके हैं?
A menu has (5) snacks and (4) drinks. In how many ways can one snack and one drink be chosen?
#permutations and combinations
#counting principle
#multiplication rule
A (20)
B (9)
C (16)
D (25)
Explanation opens after your attempt
Step 1
Concept
Total ways are \(5\times4=20\). Remember to multiply independent choices.
Step 2
Why this answer is correct
The correct answer is A. (20). Total ways are \(5\times4=20\). Remember to multiply independent choices.
Step 3
Exam Tip
कुल तरीके \(5\times4=20\) होंगे। स्वतंत्र चुनावों को गुणा करना याद रखें।
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एक दुकान में (6) प्रकार की पेंसिल और (2) प्रकार की कॉपी हैं। एक पेंसिल और एक कॉपी कितने तरीकों से खरीदी जा सकती है?
A shop has (6) types of pencils and (2) types of notebooks. In how many ways can one pencil and one notebook be bought?
#permutations and combinations
#counting principle
#multiplication rule
A (6)
B (8)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
There are two independent choices so \(6\times2=12\). Multiply the ways when steps are separate.
Step 2
Why this answer is correct
The correct answer is D. (12). There are two independent choices so \(6\times2=12\). Multiply the ways when steps are separate.
Step 3
Exam Tip
दो स्वतंत्र चुनाव हैं इसलिए \(6\times2=12\)। चरण अलग हों तो उनके तरीकों को गुणा करें।
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एक विद्यार्थी (3) विषयों में से एक और (5) खेलों में से एक चुनता है। कुल चयन कितने होंगे?
A student chooses one subject from (3) subjects and one game from (5) games. How many total selections are possible?
#permutations and combinations
#counting principle
#multiplication rule
A (8)
B (15)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
Total selections are \(3\times5=15\). In one-each selection, identify the number of stages.
Step 2
Why this answer is correct
The correct answer is B. (15). Total selections are \(3\times5=15\). In one-each selection, identify the number of stages.
Step 3
Exam Tip
कुल चयन \(3\times5=15\) हैं। “एक-एक” चयन में चरणों की संख्या पहचानें।
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एक मोबाइल लॉक में पहला अंक (1) से (5) तक और दूसरा अंक (0) से (9) तक हो सकता है। ऐसे (2) अंकों के कोड कितने होंगे?
In a mobile lock, the first digit can be from (1) to (5) and the second digit can be from (0) to (9). How many such (2)-digit codes are possible?
#permutations and combinations
#counting principle
#codes
A (15)
B (25)
C (50)
D (100)
Explanation opens after your attempt
Step 1
Concept
The first place has (5) choices and the second has (10), so \(5\times10=50\). Count choices for each position separately.
Step 2
Why this answer is correct
The correct answer is C. (50). The first place has (5) choices and the second has (10), so \(5\times10=50\). Count choices for each position separately.
Step 3
Exam Tip
पहले स्थान के (5) और दूसरे के (10) विकल्प हैं इसलिए \(5\times10=50\)। हर स्थान के विकल्प अलग गिनें।
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किसी शहर से गाँव जाने के लिए (4) बस मार्ग और (3) ट्रेन मार्ग हैं। केवल एक साधन से यात्रा कितने तरीकों से की जा सकती है?
There are (4) bus routes and (3) train routes from a city to a village. In how many ways can the trip be made using only one mode?
#permutations and combinations
#counting principle
#addition rule
A (12)
B (7)
C (4)
D (3)
Explanation opens after your attempt
Step 1
Concept
This is a bus or train choice, so (4+3=7). For mutually exclusive “or” choices, add the ways.
Step 2
Why this answer is correct
The correct answer is B. (7). This is a bus or train choice, so (4+3=7). For mutually exclusive “or” choices, add the ways.
Step 3
Exam Tip
यह बस या ट्रेन का चुनाव है इसलिए (4+3=7)। “या” में असंबद्ध विकल्प हों तो जोड़ें।
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एक परीक्षा में (6) लघु प्रश्न या (4) दीर्घ प्रश्नों में से केवल एक प्रश्न हल करना है। चयन के कितने तरीके हैं?
In an exam, exactly one question is to be solved from (6) short questions or (4) long questions. How many choices are there?
#permutations and combinations
#counting principle
#addition rule
A (10)
B (24)
C (12)
D (6)
Explanation opens after your attempt
Step 1
Concept
Only one type of question is chosen, so (6+4=10). Add the ways from separate option groups.
Step 2
Why this answer is correct
The correct answer is A. (10). Only one type of question is chosen, so (6+4=10). Add the ways from separate option groups.
Step 3
Exam Tip
केवल एक प्रकार का प्रश्न चुनना है इसलिए (6+4=10)। अलग-अलग विकल्प समूहों को जोड़ें।
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एक विद्यालय में (7) लड़के और (5) लड़कियाँ भाषण के लिए उपलब्ध हैं। एक वक्ता कितने तरीकों से चुना जा सकता है?
In a school, (7) boys and (5) girls are available for a speech. In how many ways can one speaker be chosen?
#permutations and combinations
#counting principle
#addition rule
A (35)
B (2)
C (12)
D (24)
Explanation opens after your attempt
Step 1
Concept
Only one speaker is to be chosen, so total ways are (7+5=12). Add when choosing one object from groups.
Step 2
Why this answer is correct
The correct answer is C. (12). Only one speaker is to be chosen, so total ways are (7+5=12). Add when choosing one object from groups.
Step 3
Exam Tip
एक ही वक्ता चुनना है इसलिए कुल (7+5=12)। एक वस्तु समूहों से चुननी हो तो जोड़ें।
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एक पुस्तकालय में (8) गणित की और (6) विज्ञान की पुस्तकें हैं। एक पुस्तक चुनने के कितने तरीके हैं?
A library has (8) mathematics books and (6) science books. In how many ways can one book be chosen?
#permutations and combinations
#counting principle
#addition rule
A (48)
B (14)
C (8)
D (6)
Explanation opens after your attempt
Step 1
Concept
One book is chosen from mathematics or science, so (8+6=14). For “only one,” look for the addition rule.
Step 2
Why this answer is correct
The correct answer is B. (14). One book is chosen from mathematics or science, so (8+6=14). For “only one,” look for the addition rule.
Step 3
Exam Tip
एक पुस्तक गणित या विज्ञान से चुनी जाएगी इसलिए (8+6=14)। “केवल एक” में जोड़ने का नियम देखें।
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एक यात्री शहर (A) से (B) जाने के (3) तरीके और (B) से (C) जाने के (4) तरीके रखता है। (A) से (C) जाने के कुल तरीके कितने हैं?
A traveler has (3) ways to go from city (A) to (B) and (4) ways to go from (B) to (C). How many ways are there to go from (A) to (C)?
#permutations and combinations
#counting principle
#routes
A (7)
B (9)
C (16)
D (12)
Explanation opens after your attempt
Step 1
Concept
The journey has two consecutive stages, so \(3\times4=12\). Apply multiplication for consecutive stages.
Step 2
Why this answer is correct
The correct answer is D. (12). The journey has two consecutive stages, so \(3\times4=12\). Apply multiplication for consecutive stages.
Step 3
Exam Tip
यात्रा दो लगातार चरणों में है इसलिए \(3\times4=12\)। लगातार चरणों में गुणन सिद्धांत लगाएँ।
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एक भोजन थाली में (2) रोटी विकल्प, (3) सब्ज़ी विकल्प और (2) मिठाई विकल्प हैं। एक-एक विकल्प चुनकर थाली कितने तरीकों से बनेगी?
A meal plate has (2) bread options, (3) vegetable options, and (2) dessert options. In how many ways can a plate be made by choosing one of each?
#permutations and combinations
#counting principle
#multiplication rule
A (12)
B (7)
C (10)
D (6)
Explanation opens after your attempt
Step 1
Concept
Total ways are \(2\times3\times2=12\). For three independent choices, multiply all three.
Step 2
Why this answer is correct
The correct answer is A. (12). Total ways are \(2\times3\times2=12\). For three independent choices, multiply all three.
Step 3
Exam Tip
कुल तरीके \(2\times3\times2=12\) हैं। तीन स्वतंत्र चुनाव हों तो तीनों को गुणा करें।
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एक बैग में (3) रंगों के पेन, (4) रंगों के मार्कर और (2) रंगों की पेंसिल हैं। एक पेन, एक मार्कर और एक पेंसिल कितने तरीकों से चुने जा सकते हैं?
A bag has pens in (3) colors, markers in (4) colors, and pencils in (2) colors. In how many ways can one pen, one marker, and one pencil be chosen?
#permutations and combinations
#counting principle
#multiplication rule
A (9)
B (24)
C (12)
D (18)
Explanation opens after your attempt
Step 1
Concept
The choices are independent, so \(3\times4\times2=24\). Multiply stages connected by “and.”
Step 2
Why this answer is correct
The correct answer is B. (24). The choices are independent, so \(3\times4\times2=24\). Multiply stages connected by “and.”
Step 3
Exam Tip
चुनाव स्वतंत्र हैं इसलिए \(3\times4\times2=24\)। “और” से जुड़े चरणों में गुणा करें।
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एक ऑनलाइन फॉर्म में (4) कक्षा विकल्प, (3) सेक्शन विकल्प और (5) विषय विकल्प हैं। एक पूरा चयन कितने तरीकों से होगा?
An online form has (4) class options, (3) section options, and (5) subject options. In how many ways can a complete selection be made?
#permutations and combinations
#counting principle
#multiplication rule
A (35)
B (45)
C (60)
D (12)
Explanation opens after your attempt
Step 1
Concept
Total ways are \(4\times3\times5=60\). Multiply the choices in each dropdown.
Step 2
Why this answer is correct
The correct answer is C. (60). Total ways are \(4\times3\times5=60\). Multiply the choices in each dropdown.
Step 3
Exam Tip
कुल तरीके \(4\times3\times5=60\) हैं। प्रत्येक ड्रॉपडाउन के विकल्पों को गुणा करें।
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एक उपहार पैक में (5) कवर, (2) रिबन और (3) कार्ड विकल्प हैं। एक पूरा पैक कितने तरीकों से बन सकता है?
A gift pack has (5) cover options, (2) ribbon options, and (3) card options. In how many ways can one complete pack be made?
#permutations and combinations
#counting principle
#multiplication rule
A (10)
B (20)
C (25)
D (30)
Explanation opens after your attempt
Step 1
Concept
All three choices are needed, so \(5\times2\times3=30\). Count every stage in a complete arrangement.
Step 2
Why this answer is correct
The correct answer is D. (30). All three choices are needed, so \(5\times2\times3=30\). Count every stage in a complete arrangement.
Step 3
Exam Tip
तीनों विकल्प एक साथ चाहिए इसलिए \(5\times2\times3=30\)। पूर्ण व्यवस्था में सभी चरण गिनें।
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एक (3)-अंकीय कोड में प्रत्येक स्थान पर (0) से (9) तक कोई भी अंक हो सकता है। यदि पुनरावृत्ति की अनुमति है तो कोड कितने होंगे?
In a (3)-digit code, each position can have any digit from (0) to (9). If repetition is allowed, how many codes are possible?
#permutations and combinations
#counting principle
#codes
A (1000)
B (300)
C (30)
D (999)
Explanation opens after your attempt
Step 1
Concept
Each position has (10) choices, so \(10\times10\times10=1000\). In a code, if leading (0) is allowed, take (10) choices.
Step 2
Why this answer is correct
The correct answer is A. (1000). Each position has (10) choices, so \(10\times10\times10=1000\). In a code, if leading (0) is allowed, take (10) choices.
Step 3
Exam Tip
हर स्थान के (10) विकल्प हैं इसलिए \(10\times10\times10=1000\)। कोड में पहला (0) भी चल सकता है तो (10) विकल्प लें।
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(1) से (9) तक के अंकों से पुनरावृत्ति सहित (2)-अंकीय पासकोड कितने बनेंगे?
How many (2)-digit passcodes can be formed from digits (1) to (9) with repetition allowed?
#permutations and combinations
#counting principle
#codes
A (18)
B (81)
C (72)
D (90)
Explanation opens after your attempt
Step 1
Concept
Both positions have (9) choices, so \(9\times9=81\). With repetition, choices do not decrease.
Step 2
Why this answer is correct
The correct answer is B. (81). Both positions have (9) choices, so \(9\times9=81\). With repetition, choices do not decrease.
Step 3
Exam Tip
दोनों स्थानों के (9) विकल्प हैं इसलिए \(9\times9=81\)। पुनरावृत्ति हो तो विकल्प घटते नहीं हैं।
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अंकों (2,4,6,8) से पुनरावृत्ति बिना (2)-अंकीय संख्याएँ कितनी बनेंगी?
How many (2)-digit numbers can be formed from digits (2,4,6,8) without repetition?
#permutations and combinations
#counting principle
#digits
A (8)
B (10)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
The first place has (4) choices and the second has (3), so \(4\times3=12\). Without repetition, choices decrease at the next place.
Step 2
Why this answer is correct
The correct answer is C. (12). The first place has (4) choices and the second has (3), so \(4\times3=12\). Without repetition, choices decrease at the next place.
Step 3
Exam Tip
पहले स्थान के (4) और दूसरे के (3) विकल्प हैं इसलिए \(4\times3=12\)। बिना पुनरावृत्ति हर अगले स्थान पर विकल्प घटते हैं।
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अंकों (0,1,2,3,4) से (2)-अंकीय संख्याएँ कितनी बनेंगी यदि पुनरावृत्ति की अनुमति है?
How many (2)-digit numbers can be formed from digits (0,1,2,3,4) if repetition is allowed?
#permutations and combinations
#counting principle
#digits
A (25)
B (16)
C (10)
D (20)
Explanation opens after your attempt
Step 1
Concept
The tens place cannot be (0), so it has (4) choices and the units place has (5). Total ways are \(4\times5=20\).
Step 2
Why this answer is correct
The correct answer is D. (20). The tens place cannot be (0), so it has (4) choices and the units place has (5). Total ways are \(4\times5=20\).
Step 3
Exam Tip
दहाई स्थान पर (0) नहीं आ सकता इसलिए (4) विकल्प और इकाई पर (5) विकल्प हैं। कुल \(4\times5=20\) हैं।
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अंकों (1,3,5,7,9) से पुनरावृत्ति बिना (3)-अंकीय संख्याएँ कितनी बनेंगी?
How many (3)-digit numbers can be formed from digits (1,3,5,7,9) without repetition?
#permutations and combinations
#counting principle
#digits
A (60)
B (125)
C (15)
D (30)
Explanation opens after your attempt
Step 1
Concept
The choices for three places are (5,4,3), so \(5\times4\times3=60\). While filling positions, track remaining choices.
Step 2
Why this answer is correct
The correct answer is A. (60). The choices for three places are (5,4,3), so \(5\times4\times3=60\). While filling positions, track remaining choices.
Step 3
Exam Tip
तीन स्थानों के विकल्प (5,4,3) हैं इसलिए \(5\times4\times3=60\)। स्थान भरते समय बचे हुए विकल्प देखें।
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अंकों (0,2,5,7) से पुनरावृत्ति बिना (3)-अंकीय संख्याएँ कितनी बनेंगी?
How many (3)-digit numbers can be formed from digits (0,2,5,7) without repetition?
#permutations and combinations
#counting principle
#digits
A (18)
B (24)
C (12)
D (36)
Explanation opens after your attempt
Step 1
Concept
The hundreds place cannot be (0), so it has (3) choices. Then (3) and (2) choices remain, giving \(3\times3\times2=18\).
Step 2
Why this answer is correct
The correct answer is A. (18). The hundreds place cannot be (0), so it has (3) choices. Then (3) and (2) choices remain, giving \(3\times3\times2=18\).
Step 3
Exam Tip
सैकड़ा स्थान पर (0) नहीं होगा इसलिए (3) विकल्प हैं। फिर (3) और (2) विकल्प बचते हैं इसलिए \(3\times3\times2=18\)।
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अंकों (2,3,4,5) से पुनरावृत्ति सहित (3)-अंकीय संख्याएँ कितनी बनेंगी?
How many (3)-digit numbers can be formed from digits (2,3,4,5) with repetition allowed?
#permutations and combinations
#counting principle
#digits
A (12)
B (64)
C (24)
D (48)
Explanation opens after your attempt
Step 1
Concept
Each place has (4) choices, so \(4\times4\times4=64\). With repetition, each place keeps the same choices.
Step 2
Why this answer is correct
The correct answer is B. (64). Each place has (4) choices, so \(4\times4\times4=64\). With repetition, each place keeps the same choices.
Step 3
Exam Tip
हर स्थान के (4) विकल्प हैं इसलिए \(4\times4\times4=64\)। पुनरावृत्ति होने पर हर स्थान पर समान विकल्प रहते हैं।
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अंकों (1,2,3,4,5,6) से पुनरावृत्ति बिना (2)-अंकीय सम संख्याएँ कितनी बनेंगी?
How many (2)-digit even numbers can be formed from digits (1,2,3,4,5,6) without repetition?
#permutations and combinations
#counting principle
#even numbers
A (12)
B (18)
C (15)
D (30)
Explanation opens after your attempt
Step 1
Concept
The units place has (3) choices from (2,4,6), and (5) choices remain for the tens place. Total ways are \(3\times5=15\).
Step 2
Why this answer is correct
The correct answer is C. (15). The units place has (3) choices from (2,4,6), and (5) choices remain for the tens place. Total ways are \(3\times5=15\).
Step 3
Exam Tip
इकाई स्थान पर (2,4,6) में से (3) विकल्प हैं और दहाई पर (5) विकल्प बचते हैं। कुल \(3\times5=15\) हैं।
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अंकों (0,1,2,3,4,5) से पुनरावृत्ति सहित (2)-अंकीय विषम संख्याएँ कितनी बनेंगी?
How many (2)-digit odd numbers can be formed from digits (0,1,2,3,4,5) with repetition allowed?
#permutations and combinations
#counting principle
#odd numbers
A (30)
B (18)
C (12)
D (15)
Explanation opens after your attempt
Step 1
Concept
The units place has (3) choices (1,3,5), and the tens place has (5) choices from (1) to (5). Total ways are \(5\times3=15\).
Step 2
Why this answer is correct
The correct answer is D. (15). The units place has (3) choices (1,3,5), and the tens place has (5) choices from (1) to (5). Total ways are \(5\times3=15\).
Step 3
Exam Tip
इकाई स्थान पर (1,3,5) के (3) विकल्प हैं और दहाई पर (1) से (5) तक (5) विकल्प हैं। कुल \(5\times3=15\) हैं।
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एक (2)-अक्षर का कोड अंग्रेज़ी के (5) दिए गए अक्षरों से बनाना है। यदि अक्षर दोहराए जा सकते हैं तो कितने कोड बनेंगे?
A (2)-letter code is to be formed from (5) given English letters. If letters may be repeated, how many codes are possible?
#permutations and combinations
#counting principle
#codes
A (25)
B (10)
C (20)
D (15)
Explanation opens after your attempt
Step 1
Concept
Each position can take (5) letters, so \(5\times5=25\). If repetition is allowed in a code, keep the same choices for each position.
Step 2
Why this answer is correct
The correct answer is A. (25). Each position can take (5) letters, so \(5\times5=25\). If repetition is allowed in a code, keep the same choices for each position.
Step 3
Exam Tip
हर स्थान पर (5) अक्षर आ सकते हैं इसलिए \(5\times5=25\)। कोड में पुनरावृत्ति हो तो प्रत्येक स्थान पर समान विकल्प रखें।
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एक (3)-अक्षर का कोड (6) अलग अक्षरों से बनाना है। यदि पुनरावृत्ति नहीं है तो कितने कोड बनेंगे?
A (3)-letter code is to be formed from (6) distinct letters. If repetition is not allowed, how many codes are possible?
#permutations and combinations
#counting principle
#codes
A (216)
B (120)
C (18)
D (36)
Explanation opens after your attempt
Step 1
Concept
For three positions, choices are (6,5,4), so \(6\times5\times4=120\). Without repetition, a chosen letter is no longer available.
Step 2
Why this answer is correct
The correct answer is B. (120). For three positions, choices are (6,5,4), so \(6\times5\times4=120\). Without repetition, a chosen letter is no longer available.
Step 3
Exam Tip
तीन स्थानों के लिए (6,5,4) विकल्प हैं इसलिए \(6\times5\times4=120\)। बिना पुनरावृत्ति चुना हुआ अक्षर फिर उपलब्ध नहीं रहता।
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एक वाहन नंबर में (2) अक्षर और उसके बाद (2) अंक हैं। अक्षरों के (4) विकल्प और अंकों के (10) विकल्प हैं। पुनरावृत्ति की अनुमति है तो कुल नंबर कितने बनेंगे?
A vehicle number has (2) letters followed by (2) digits. There are (4) choices for each letter and (10) choices for each digit. If repetition is allowed, how many numbers can be formed?
#permutations and combinations
#counting principle
#number plate
A (160)
B (400)
C (1600)
D (1000)
Explanation opens after your attempt
Step 1
Concept
The four positions have choices (4,4,10,10). Total numbers are \(4\times4\times10\times10=1600\).
Step 2
Why this answer is correct
The correct answer is C. (1600). The four positions have choices (4,4,10,10). Total numbers are \(4\times4\times10\times10=1600\).
Step 3
Exam Tip
चार स्थानों के विकल्प (4,4,10,10) हैं। कुल \(4\times4\times10\times10=1600\) होंगे।
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