Class 11 Mathematics Hard Quiz

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शब्द (ALGEBRA) के अक्षरों को इस प्रकार कितने तरीकों से व्यवस्थित किया जा सकता है कि सभी स्वर साथ-साथ रहें?

In how many ways can the letters of the word (ALGEBRA) be arranged so that all vowels stay together?

Explanation opens after your attempt
Correct Answer

B. (360)

Step 1

Concept

Treat the vowels as one block, giving (5!) arrangements and (3!/2!) internal arrangements. In exams, always check repeated letters.

Step 2

Why this answer is correct

The correct answer is B. (360). Treat the vowels as one block, giving (5!) arrangements and (3!/2!) internal arrangements. In exams, always check repeated letters.

Step 3

Exam Tip

स्वरों को एक खंड मानने पर (5!) व्यवस्थाएं और भीतर (3!/2!) व्यवस्थाएं मिलती हैं। परीक्षा में समान अक्षरों की पुनरावृत्ति जरूर जांचें।

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शब्द (ASSESS) के अक्षरों से बनने वाली भिन्न व्यवस्थाओं की संख्या क्या है?

What is the number of distinct arrangements formed from the letters of (ASSESS)?

Explanation opens after your attempt
Correct Answer

B. (30)

Step 1

Concept

There are (6) letters with (S) repeated four times, so the number is (6!/4!). In exams, do not forget to divide by repeated letters.

Step 2

Why this answer is correct

The correct answer is B. (30). There are (6) letters with (S) repeated four times, so the number is (6!/4!). In exams, do not forget to divide by repeated letters.

Step 3

Exam Tip

कुल (6) अक्षरों में (S) चार बार है, इसलिए संख्या (6!/4!) है। परीक्षा में समान अक्षरों से भाग देना न भूलें।

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अंकों (0,1,2,3,4,5) से बिना पुनरावृत्ति कितनी (4)-अंकीय संख्याएं बनेंगी जो (5) से विभाज्य हों?

How many (4)-digit numbers divisible by (5) can be formed from digits (0,1,2,3,4,5) without repetition?

Explanation opens after your attempt
Correct Answer

A. (108)

Step 1

Concept

The last digit must be (0) or (5), and the leading zero case must be counted separately. For such questions, start with the unit digit.

Step 2

Why this answer is correct

The correct answer is A. (108). The last digit must be (0) or (5), and the leading zero case must be counted separately. For such questions, start with the unit digit.

Step 3

Exam Tip

अंतिम अंक (0) या (5) होगा और पहले अंक के शून्य होने की स्थिति अलग गिननी होगी। ऐसे प्रश्नों में अंतिम अंक से केस बनाएं।

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(7) व्यक्तियों को एक पंक्ति में कितने तरीकों से बैठाया जा सकता है यदि दो विशेष व्यक्ति साथ न बैठें?

In how many ways can (7) people be seated in a row if two particular people must not sit together?

Explanation opens after your attempt
Correct Answer

A. (3600)

Step 1

Concept

Subtract the together cases \(2\cdot 6!\) from the total (7!). For restriction questions, subtracting bad cases is often easier.

Step 2

Why this answer is correct

The correct answer is A. (3600). Subtract the together cases \(2\cdot 6!\) from the total (7!). For restriction questions, subtracting bad cases is often easier.

Step 3

Exam Tip

कुल (7!) में से साथ बैठने वाली \(2\cdot 6!\) व्यवस्थाएं घटाएं। निषेध वाले प्रश्नों में कुल से खराब व्यवस्थाएं घटाना आसान होता है।

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(8) लोगों को गोल मेज पर कितने तरीकों से बैठाया जा सकता है यदि दो विशेष लोग हमेशा साथ बैठें?

In how many ways can (8) people be seated around a circular table if two particular people always sit together?

Explanation opens after your attempt
Correct Answer

B. (1440)

Step 1

Concept

Treat the two people as one block, so (7) units have circular arrangements (6!), with (2!) internal ways. In circular arrangements, rotations are identical.

Step 2

Why this answer is correct

The correct answer is B. (1440). Treat the two people as one block, so (7) units have circular arrangements (6!), with (2!) internal ways. In circular arrangements, rotations are identical.

Step 3

Exam Tip

दो लोगों को एक खंड मानकर (7) इकाइयों की गोल व्यवस्था (6!) और भीतर (2!) तरीके हैं। गोल व्यवस्था में घूर्णन को स्थिर मानें।

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शब्द (TRIANGLE) के अक्षरों से कितनी व्यवस्थाएं बनेंगी जिनमें (T) और (E) के बीच ठीक (3) अक्षर हों?

How many arrangements of the letters of (TRIANGLE) have exactly (3) letters between (T) and (E)?

Explanation opens after your attempt
Correct Answer

C. (1920)

Step 1

Concept

There are (4) possible position pairs for (T) and (E), with (2) orders, and the remaining (6!) arrangements. In position-based questions, choose fixed positions first.

Step 2

Why this answer is correct

The correct answer is C. (1920). There are (4) possible position pairs for (T) and (E), with (2) orders, and the remaining (6!) arrangements. In position-based questions, choose fixed positions first.

Step 3

Exam Tip

(T) और (E) की जगहों के (4) संभावित जोड़े हैं और क्रम (2) तरीके से होगा, बाकी (6!) हैं। स्थान-आधारित प्रश्नों में पहले निश्चित स्थान चुनें।

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(1,2,3,4,5,6,7) अंकों से बिना पुनरावृत्ति कितनी (5)-अंकीय सम संख्याएं बन सकती हैं?

How many (5)-digit even numbers can be formed from digits (1,2,3,4,5,6,7) without repetition?

Explanation opens after your attempt
Correct Answer

B. (1080)

Step 1

Concept

The last digit can be (2,4,6) in (3) ways, and the remaining four places in \(^{6}P_{4}\) ways. For even numbers, fix the unit digit first.

Step 2

Why this answer is correct

The correct answer is B. (1080). The last digit can be (2,4,6) in (3) ways, and the remaining four places in \(^{6}P_{4}\) ways. For even numbers, fix the unit digit first.

Step 3

Exam Tip

अंतिम अंक (2,4,6) में से (3) तरीकों से होगा और शेष चार स्थान \(^{6}P_{4}\) तरीकों से भरेंगे। सम संख्या में इकाई अंक पहले तय करें।

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शब्द (COMMITTEE) के अक्षरों की भिन्न व्यवस्थाओं की संख्या कितनी है?

What is the number of distinct arrangements of the letters of (COMMITTEE)?

Explanation opens after your attempt
Correct Answer

A. (45360)

Step 1

Concept

There are (9) letters with (M,T,E) each repeated twice, so the count is (9!/(2!2!2!)). For difficult words, list letter counts first.

Step 2

Why this answer is correct

The correct answer is A. (45360). There are (9) letters with (M,T,E) each repeated twice, so the count is (9!/(2!2!2!)). For difficult words, list letter counts first.

Step 3

Exam Tip

कुल (9) अक्षरों में (M,T,E) दो-दो बार हैं, इसलिए संख्या (9!/(2!2!2!)) है। कठिन शब्दों में अक्षरों की गिनती पहले लिखें।

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(5) लड़के और (4) लड़कियां एक पंक्ति में कितने तरीकों से बैठ सकते हैं यदि कोई दो लड़कियां साथ न बैठें?

In how many ways can (5) boys and (4) girls sit in a row if no two girls sit together?

Explanation opens after your attempt
Correct Answer

A. (86400)

Step 1

Concept

Arrange the boys in (5!) ways, then place the girls in \(^{6}P_{4}\) ways in (4) of the (6) gaps. The gap method is safest for such questions.

Step 2

Why this answer is correct

The correct answer is A. (86400). Arrange the boys in (5!) ways, then place the girls in \(^{6}P_{4}\) ways in (4) of the (6) gaps. The gap method is safest for such questions.

Step 3

Exam Tip

पहले लड़कों को (5!) तरीकों से बैठाएं और (6) खाली स्थानों में से (4) पर लड़कियां \(^{6}P_{4}\) तरीकों से बैठेंगी। अंतराल विधि ऐसे प्रश्नों में सबसे सुरक्षित है।

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(6) अलग-अलग पुस्तकों को एक शेल्फ पर कितने तरीकों से लगाया जा सकता है यदि दो निर्धारित पुस्तकें सिरों पर ही हों?

In how many ways can (6) distinct books be arranged on a shelf if two specified books must occupy the end positions?

Explanation opens after your attempt
Correct Answer

B. (48)

Step 1

Concept

The two specified books occupy the ends in (2!) ways, and the remaining (4) books in (4!) ways. Fill the most restricted places first.

Step 2

Why this answer is correct

The correct answer is B. (48). The two specified books occupy the ends in (2!) ways, and the remaining (4) books in (4!) ways. Fill the most restricted places first.

Step 3

Exam Tip

दो निर्धारित पुस्तकें सिरों पर (2!) तरीकों से और शेष (4) पुस्तकें (4!) तरीकों से लगेंगी। पहले सबसे अधिक प्रतिबंधित स्थान भरें।

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अक्षरों (A,B,C,D,E,F) से बने शब्दों में (B) हमेशा (A) के बाद आए, ऐसी कितनी व्यवस्थाएं हैं?

Among arrangements of letters (A,B,C,D,E,F), how many have (B) always appearing after (A)?

Explanation opens after your attempt
Correct Answer

C. (360)

Step 1

Concept

In the total (6!) arrangements, the two orders of (A) and (B) are equally likely, so half are valid. Symmetry saves time in order-condition questions.

Step 2

Why this answer is correct

The correct answer is C. (360). In the total (6!) arrangements, the two orders of (A) and (B) are equally likely, so half are valid. Symmetry saves time in order-condition questions.

Step 3

Exam Tip

कुल (6!) व्यवस्थाओं में (A) और (B) के दोनों क्रम बराबर हैं, इसलिए आधी व्यवस्थाएं सही हैं। क्रम-शर्त में सममिति से समय बचता है।

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शब्द (EQUATION) के अक्षरों को कितने तरीकों से व्यवस्थित किया जा सकता है कि स्वर केवल विषम स्थानों पर आएं?

In how many ways can the letters of (EQUATION) be arranged so that vowels occupy only odd positions?

Explanation opens after your attempt
Correct Answer

A. (1440)

Step 1

Concept

There are (4) odd positions among (8), and the (4) vowels occupy them in (4!) ways, while consonants also arrange in (4!) ways. When counts match, arrange groups separately.

Step 2

Why this answer is correct

The correct answer is A. (1440). There are (4) odd positions among (8), and the (4) vowels occupy them in (4!) ways, while consonants also arrange in (4!) ways. When counts match, arrange groups separately.

Step 3

Exam Tip

(8) स्थानों में (4) विषम स्थान हैं और (4) स्वर उन्हीं में (4!) तरीकों से आएंगे, व्यंजन भी (4!) तरीकों से। स्थान की समान संख्या दिखे तो सीधे अलग-अलग व्यवस्थित करें।

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(0,1,2,3,4,5,6) से बिना पुनरावृत्ति कितनी (5)-अंकीय संख्याएं बनेंगी जिनमें पहला अंक (0) न हो और संख्या (3) से विभाज्य हो?

How many (5)-digit numbers divisible by (3) can be formed from (0,1,2,3,4,5,6) without repetition and with first digit not (0)?

Explanation opens after your attempt
Correct Answer

C. (360)

Step 1

Concept

The digit sum must be divisible by (3); arrange valid (5)-digit sets and subtract leading-zero arrangements. Choosing digit sets first is useful.

Step 2

Why this answer is correct

The correct answer is C. (360). The digit sum must be divisible by (3); arrange valid (5)-digit sets and subtract leading-zero arrangements. Choosing digit sets first is useful.

Step 3

Exam Tip

अंकों का योग (3) से विभाज्य होना चाहिए; मान्य (5)-अंकीय समूहों की व्यवस्थाओं में अग्र शून्य घटाएं। पहले अंकों के समूह चुनना उपयोगी है।

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(9) अलग-अलग मोतियों की माला कितने तरीकों से बनाई जा सकती है यदि घुमाने और पलटने पर मालाएं समान मानी जाएं?

In how many ways can a necklace be made using (9) distinct beads if rotations and reflections are considered identical?

Explanation opens after your attempt
Correct Answer

A. (20160)

Step 1

Concept

For a necklace, the count is ((9-1)!/2) because both rotation and reflection are identical. Remember the difference between circular seating and necklaces.

Step 2

Why this answer is correct

The correct answer is A. (20160). For a necklace, the count is ((9-1)!/2) because both rotation and reflection are identical. Remember the difference between circular seating and necklaces.

Step 3

Exam Tip

माला में संख्या ((9-1)!/2) होती है क्योंकि घूर्णन और प्रतिबिंब दोनों समान माने जाते हैं। गोल मेज और माला के अंतर को याद रखें।

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शब्द (MOTHER) के अक्षरों की शब्दकोशीय क्रम में रैंक क्या होगी?

What is the lexicographic rank of the word (MOTHER) among arrangements of its letters?

Explanation opens after your attempt
Correct Answer

B. (360)

Step 1

Concept

Add arrangements beginning with smaller available letters at each position, then add (1). In rank questions, write the letters in alphabetical order first.

Step 2

Why this answer is correct

The correct answer is B. (360). Add arrangements beginning with smaller available letters at each position, then add (1). In rank questions, write the letters in alphabetical order first.

Step 3

Exam Tip

हर स्थान पर उससे छोटे उपलब्ध अक्षरों से शुरू होने वाली व्यवस्थाएं जोड़कर अंत में (1) जोड़ते हैं। रैंक प्रश्नों में अक्षरों को पहले वर्णक्रम में लिखें।

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(4) पत्रों को (4) सही लिफाफों में इस प्रकार डालने के तरीकों की संख्या कितनी है कि कोई भी पत्र सही लिफाफे में न जाए?

How many ways are there to put (4) letters into (4) addressed envelopes so that no letter goes into the correct envelope?

Explanation opens after your attempt
Correct Answer

C. (9)

Step 1

Concept

This is the derangement of (4) objects, whose value is (9). For such questions, use the derangement formula or a small listing.

Step 2

Why this answer is correct

The correct answer is C. (9). This is the derangement of (4) objects, whose value is (9). For such questions, use the derangement formula or a small listing.

Step 3

Exam Tip

यह (4) वस्तुओं का पूर्ण विस्थापन है और मान (9) होता है। ऐसे प्रश्नों में विस्थापन सूत्र या छोटी सूची काम आती है।

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(10) धावकों की दौड़ में प्रथम (3) स्थानों के परिणाम कितने तरीकों से हो सकते हैं?

In a race with (10) runners, in how many ways can the first (3) positions be decided?

Explanation opens after your attempt
Correct Answer

C. (720)

Step 1

Concept

This is ordered selection, so \(^{10}P_{3}=10\cdot 9\cdot 8\). When positions are ranked, order matters.

Step 2

Why this answer is correct

The correct answer is C. (720). This is ordered selection, so \(^{10}P_{3}=10\cdot 9\cdot 8\). When positions are ranked, order matters.

Step 3

Exam Tip

यह क्रम सहित चयन है, इसलिए \(^{10}P_{3}=10\cdot 9\cdot 8\) होगा। स्थानों की रैंकिंग हो तो क्रम महत्वपूर्ण होता है।

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(8) लोगों में से अध्यक्ष, सचिव और कोषाध्यक्ष कितने तरीकों से चुने जा सकते हैं यदि कोई व्यक्ति एक से अधिक पद न ले?

From (8) people, in how many ways can a president, secretary and treasurer be chosen if no person holds more than one post?

Explanation opens after your attempt
Correct Answer

A. (336)

Step 1

Concept

The posts are distinct, so the number of ways is \(^{8}P_{3}\). Choosing distinct posts is an ordered selection.

Step 2

Why this answer is correct

The correct answer is A. (336). The posts are distinct, so the number of ways is \(^{8}P_{3}\). Choosing distinct posts is an ordered selection.

Step 3

Exam Tip

पद अलग-अलग हैं, इसलिए \(^{8}P_{3}\) तरीके होंगे। अलग पदों का चयन क्रम सहित होता है।

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शब्द (BALLOON) के अक्षरों की कितनी भिन्न व्यवस्थाएं बनेंगी?

How many distinct arrangements can be formed from the letters of (BALLOON)?

Explanation opens after your attempt
Correct Answer

A. (1260)

Step 1

Concept

There are (7) letters with (L) and (O) each repeated twice, so the count is (7!/(2!2!)). Repeated letters are not treated as distinct objects.

Step 2

Why this answer is correct

The correct answer is A. (1260). There are (7) letters with (L) and (O) each repeated twice, so the count is (7!/(2!2!)). Repeated letters are not treated as distinct objects.

Step 3

Exam Tip

(7) अक्षरों में (L) और (O) दो-दो बार हैं, इसलिए संख्या (7!/(2!2!)) है। समान अक्षर हर बार अलग वस्तु नहीं माने जाते।

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शब्द (BALLOON) की व्यवस्थाओं में दोनों (L) साथ हों, ऐसी कितनी भिन्न व्यवस्थाएं हैं?

In arrangements of (BALLOON), how many distinct arrangements have both (L)'s together?

Explanation opens after your attempt
Correct Answer

B. (360)

Step 1

Concept

Treat the two (L)'s as one block, giving (6) units with (O) repeated twice, so the count is (6!/2!). Use the block method for together conditions.

Step 2

Why this answer is correct

The correct answer is B. (360). Treat the two (L)'s as one block, giving (6) units with (O) repeated twice, so the count is (6!/2!). Use the block method for together conditions.

Step 3

Exam Tip

दोनों (L) को एक खंड मानने पर (6) इकाइयां हैं और (O) दो बार है, इसलिए (6!/2!) है। साथ वाली शर्त में खंड विधि लगाएं।

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अंकों (1,2,3,4,5,6,7,8) से बिना पुनरावृत्ति कितनी (4)-अंकीय संख्याएं (4000) से बड़ी बनेंगी?

How many (4)-digit numbers greater than (4000) can be formed from (1,2,3,4,5,6,7,8) without repetition?

Explanation opens after your attempt
Correct Answer

A. (840)

Step 1

Concept

The thousands digit can be (4,5,6,7,8), and the remaining (3) places are filled in \(^{7}P_{3}\) ways. In comparison questions, fix the first digit first.

Step 2

Why this answer is correct

The correct answer is A. (840). The thousands digit can be (4,5,6,7,8), and the remaining (3) places are filled in \(^{7}P_{3}\) ways. In comparison questions, fix the first digit first.

Step 3

Exam Tip

हजार का अंक (4,5,6,7,8) में से होगा और बाकी (3) स्थान \(^{7}P_{3}\) तरीकों से भरेंगे। तुलना वाले प्रश्नों में पहले पहला अंक तय करें।

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(6) पुरुष और (6) महिलाएं एक पंक्ति में कितने तरीकों से बैठ सकते हैं यदि वे बारी-बारी से बैठें?

In how many ways can (6) men and (6) women sit in a row if they sit alternately?

Explanation opens after your attempt
Correct Answer

A. \(2\cdot 6!\cdot 6!\)

Step 1

Concept

There are two possible patterns, and in each pattern men arrange in (6!) ways and women in (6!) ways. For equal alternate seating, remember the two patterns.

Step 2

Why this answer is correct

The correct answer is A. \(2\cdot 6!\cdot 6!\). There are two possible patterns, and in each pattern men arrange in (6!) ways and women in (6!) ways. For equal alternate seating, remember the two patterns.

Step 3

Exam Tip

बैठने की दो रूपरेखाएं संभव हैं और हर रूपरेखा में पुरुष (6!) तथा महिलाएं (6!) तरीकों से बैठते हैं। समान संख्या में वैकल्पिक बैठाने पर दो पैटर्न याद रखें।

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शब्द (PERMUTE) के अक्षरों से कितने शब्द बनेंगे जिनमें सभी स्वर वर्णक्रम में रहें?

How many arrangements of (PERMUTE) have all vowels in alphabetical order?

Explanation opens after your attempt
Correct Answer

B. (840)

Step 1

Concept

The total distinct arrangements are (7!/2!), and among the possible vowel orders only one valid alphabetical order is allowed. Count order restrictions carefully when vowels repeat.

Step 2

Why this answer is correct

The correct answer is B. (840). The total distinct arrangements are (7!/2!), and among the possible vowel orders only one valid alphabetical order is allowed. Count order restrictions carefully when vowels repeat.

Step 3

Exam Tip

कुल (7!) व्यवस्थाओं में (E,E,U) के प्रभाव से भिन्न व्यवस्थाएं (7!/2!) हैं और स्वरों के क्रमों में केवल एक क्रम मान्य है। समान स्वरों सहित क्रम-शर्त सावधानी से गिनें।

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(5) अलग-अलग चाबियों को (5) अलग-अलग ताले में आजमाने के क्रम कितने हैं यदि हर चाबी ठीक एक बार आजमाई जाए?

How many orders are possible for trying (5) distinct keys in (5) distinct locks if each key is tried exactly once?

Explanation opens after your attempt
Correct Answer

C. (120)

Step 1

Concept

The orders of (5) distinct objects are (5!). When only order is asked, apply factorial directly.

Step 2

Why this answer is correct

The correct answer is C. (120). The orders of (5) distinct objects are (5!). When only order is asked, apply factorial directly.

Step 3

Exam Tip

(5) अलग वस्तुओं के क्रम (5!) होते हैं। केवल क्रम पूछा हो तो सीधा फैक्टोरियल लगाएं।

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शब्द (SCHOOL) के अक्षरों की कितनी व्यवस्थाएं होंगी जिनमें दोनों (O) साथ न हों?

How many arrangements of (SCHOOL) have the two (O)'s not together?

Explanation opens after your attempt
Correct Answer

A. (240)

Step 1

Concept

Subtract the together cases (5!) from the total (6!/2!). For not-together conditions, subtract together arrangements from total.

Step 2

Why this answer is correct

The correct answer is A. (240). Subtract the together cases (5!) from the total (6!/2!). For not-together conditions, subtract together arrangements from total.

Step 3

Exam Tip

कुल (6!/2!) में से (O) साथ वाली (5!) व्यवस्थाएं घटाएं। न-साथ वाली शर्त में कुल से साथ वाली व्यवस्थाएं घटाएं।

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(7) अलग-अलग झंडों को एक ऊर्ध्वाधर डंडे पर कितने क्रमों में लगाया जा सकता है यदि एक विशेष झंडा सबसे ऊपर न हो?

In how many orders can (7) distinct flags be placed on a vertical pole if one specified flag is not at the top?

Explanation opens after your attempt
Correct Answer

B. (4320)

Step 1

Concept

Subtract the (6!) arrangements where the specified flag is at the top from the total (7!). For one forbidden position, subtraction is quick.

Step 2

Why this answer is correct

The correct answer is B. (4320). Subtract the (6!) arrangements where the specified flag is at the top from the total (7!). For one forbidden position, subtraction is quick.

Step 3

Exam Tip

कुल (7!) में से विशेष झंडे के ऊपर होने पर (6!) व्यवस्थाएं घटाएं। निषेध वाली एक जगह हो तो घटाने की विधि तेज होती है।

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(0,1,2,3,4,5) अंकों से बिना पुनरावृत्ति कितनी (4)-अंकीय विषम संख्याएं बन सकती हैं?

How many (4)-digit odd numbers can be formed from digits (0,1,2,3,4,5) without repetition?

Explanation opens after your attempt
Correct Answer

B. (144)

Step 1

Concept

The last digit can be (1,3,5) in (3) ways; the first place then has (4) nonzero choices, followed by \(4\cdot 3\). For odd numbers, fix the unit digit first.

Step 2

Why this answer is correct

The correct answer is B. (144). The last digit can be (1,3,5) in (3) ways; the first place then has (4) nonzero choices, followed by \(4\cdot 3\). For odd numbers, fix the unit digit first.

Step 3

Exam Tip

अंतिम अंक (1,3,5) में से (3) तरीकों से होगा; पहले स्थान के लिए शून्य को छोड़कर (4) विकल्प और बाकी \(4\cdot 3\) हैं। विषम संख्या में इकाई अंक पहले तय करें।

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शब्द (STATISTICS) के अक्षरों की भिन्न व्यवस्थाओं की संख्या कितनी है?

What is the number of distinct arrangements of the letters of (STATISTICS)?

Explanation opens after your attempt
Correct Answer

A. (50400)

Step 1

Concept

There are (10) letters with (S) three times, (T) three times and (I) twice, so the count is (10!/(3!3!2!)). For long words, make a frequency table.

Step 2

Why this answer is correct

The correct answer is A. (50400). There are (10) letters with (S) three times, (T) three times and (I) twice, so the count is (10!/(3!3!2!)). For long words, make a frequency table.

Step 3

Exam Tip

(10) अक्षरों में (S) तीन बार, (T) तीन बार और (I) दो बार है, इसलिए संख्या (10!/(3!3!2!)) है। बड़े शब्दों में आवृत्ति तालिका बनाएं।

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(6) विद्यार्थियों को (3) अलग-अलग पुरस्कार दिए जाने हैं और कोई विद्यार्थी एक से अधिक पुरस्कार नहीं पाएगा। कितने तरीके संभव हैं?

(3) different prizes are to be given to (6) students, and no student can receive more than one prize. How many ways are possible?

Explanation opens after your attempt
Correct Answer

C. (120)

Step 1

Concept

The prizes are distinct, so the number of ways is \(^{6}P_{3}\). For different prizes, order matters.

Step 2

Why this answer is correct

The correct answer is C. (120). The prizes are distinct, so the number of ways is \(^{6}P_{3}\). For different prizes, order matters.

Step 3

Exam Tip

पुरस्कार अलग-अलग हैं, इसलिए \(^{6}P_{3}\) तरीके होंगे। अलग पुरस्कारों में क्रम का महत्व होता है।

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(8) अलग-अलग व्यक्तियों में से (5) को एक पंक्ति में बैठाने के कितने तरीके हैं?

In how many ways can (5) out of (8) distinct persons be seated in a row?

Explanation opens after your attempt
Correct Answer

B. (6720)

Step 1

Concept

We must choose and arrange (5) persons, so the number is \(^{8}P_{5}\). Seating includes both selection and order.

Step 2

Why this answer is correct

The correct answer is B. (6720). We must choose and arrange (5) persons, so the number is \(^{8}P_{5}\). Seating includes both selection and order.

Step 3

Exam Tip

पहले (5) व्यक्तियों को क्रम सहित चुनना है, इसलिए \(^{8}P_{5}\) होगा। बैठाने में चयन और क्रम दोनों शामिल हैं।

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शब्द (NUMBER) के अक्षरों की व्यवस्थाओं में (N) और (R) सिरों पर हों, ऐसी कितनी व्यवस्थाएं हैं?

In arrangements of the letters of (NUMBER), how many have (N) and (R) at the two ends?

Explanation opens after your attempt
Correct Answer

B. (48)

Step 1

Concept

(N) and (R) can occupy the ends in (2!) ways, and the remaining (4) letters in (4!) ways. With end restrictions, fill the ends first.

Step 2

Why this answer is correct

The correct answer is B. (48). (N) and (R) can occupy the ends in (2!) ways, and the remaining (4) letters in (4!) ways. With end restrictions, fill the ends first.

Step 3

Exam Tip

(N) और (R) सिरों पर (2!) तरीकों से होंगे और शेष (4) अक्षर (4!) तरीकों से। सिरों पर शर्त हो तो पहले सिरों को भरें।

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कितनी (6)-अंकीय संख्याएं (1,2,3,4,5,6) से बिना पुनरावृत्ति बनेंगी जिनमें (1) और (2) साथ हों?

How many (6)-digit numbers can be formed from (1,2,3,4,5,6) without repetition if (1) and (2) are adjacent?

Explanation opens after your attempt
Correct Answer

C. (240)

Step 1

Concept

Treat (1) and (2) as a block, giving (5!) arrangements and (2!) internal orders. For adjacent digits, the block method is direct.

Step 2

Why this answer is correct

The correct answer is C. (240). Treat (1) and (2) as a block, giving (5!) arrangements and (2!) internal orders. For adjacent digits, the block method is direct.

Step 3

Exam Tip

(1) और (2) को खंड मानकर (5!) व्यवस्थाएं और खंड के भीतर (2!) क्रम हैं। साथ वाले अंकों में खंड विधि सीधी होती है।

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(9) अलग-अलग पुस्तकों में से (4) पुस्तकों को शेल्फ पर कितने तरीकों से लगाया जा सकता है?

In how many ways can (4) books be arranged on a shelf from (9) distinct books?

Explanation opens after your attempt
Correct Answer

A. (3024)

Step 1

Concept

This is an ordered selection of (4) books from (9), so the number is \(^{9}P_{4}\). Arrangement on a shelf requires order.

Step 2

Why this answer is correct

The correct answer is A. (3024). This is an ordered selection of (4) books from (9), so the number is \(^{9}P_{4}\). Arrangement on a shelf requires order.

Step 3

Exam Tip

यह (9) में से (4) पुस्तकों का क्रम सहित चयन है, इसलिए \(^{9}P_{4}\) होगा। शेल्फ पर लगाने में क्रम जरूरी है।

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शब्द (DELHI) के अक्षरों से बनने वाले शब्दों की वर्णक्रमीय सूची में (HIELD) की रैंक क्या है?

What is the rank of (HIELD) in the alphabetical list of words formed from the letters of (DELHI)?

Explanation opens after your attempt
Correct Answer

C. (84)

Step 1

Concept

Adding arrangements starting with smaller available letters at each position and then adding (1) gives rank (84). In rank problems, count position by position.

Step 2

Why this answer is correct

The correct answer is C. (84). Adding arrangements starting with smaller available letters at each position and then adding (1) gives rank (84). In rank problems, count position by position.

Step 3

Exam Tip

हर स्थान पर छोटे उपलब्ध अक्षरों से शुरू होने वाली व्यवस्थाएं जोड़कर (1) जोड़ने पर रैंक (84) मिलती है। रैंक में शुरुआत से अंत तक स्थानवार गिनें।

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(8) लोगों को गोल मेज पर कितने तरीकों से बैठाया जा सकता है यदि दो विशेष व्यक्ति साथ न बैठें?

In how many ways can (8) people be seated around a circular table if two particular people are not adjacent?

Explanation opens after your attempt
Correct Answer

A. (3600)

Step 1

Concept

Total circular arrangements are (7!), and adjacent cases are \(2\cdot 6!\), so subtraction gives (3600). In circular arrangements, start with ((n-1)!).

Step 2

Why this answer is correct

The correct answer is A. (3600). Total circular arrangements are (7!), and adjacent cases are \(2\cdot 6!\), so subtraction gives (3600). In circular arrangements, start with ((n-1)!).

Step 3

Exam Tip

कुल गोल व्यवस्थाएं (7!) हैं और साथ वाली \(2\cdot 6!\) हैं, इसलिए घटाने पर (3600) मिलता है। गोल व्यवस्था में कुल ((n-1)!) से शुरू करें।

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शब्द (CALCULUS) की कितनी भिन्न व्यवस्थाएं बनेंगी?

How many distinct arrangements can be formed from the letters of (CALCULUS)?

Explanation opens after your attempt
Correct Answer

A. (10080)

Step 1

Concept

There are (8) letters with (C,L,U) each repeated twice, so the count is (8!/(2!2!2!)). Identifying repetitions is the key step.

Step 2

Why this answer is correct

The correct answer is A. (10080). There are (8) letters with (C,L,U) each repeated twice, so the count is (8!/(2!2!2!)). Identifying repetitions is the key step.

Step 3

Exam Tip

(8) अक्षरों में (C,L,U) दो-दो बार हैं, इसलिए संख्या (8!/(2!2!2!)) है। आवृत्तियों को पहचानना मुख्य कदम है।

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(7) अक्षरों (A,B,C,D,E,F,G) से बनने वाली व्यवस्थाओं में (A) और (B) के बीच ठीक (2) अक्षर हों, ऐसी कितनी व्यवस्थाएं हैं?

In arrangements of (A,B,C,D,E,F,G), how many have exactly (2) letters between (A) and (B)?

Explanation opens after your attempt
Correct Answer

B. (960)

Step 1

Concept

There are (4) position pairs for (A,B) and (2) orders, then (5!) ways for the rest. For a fixed gap, count position pairs.

Step 2

Why this answer is correct

The correct answer is B. (960). There are (4) position pairs for (A,B) and (2) orders, then (5!) ways for the rest. For a fixed gap, count position pairs.

Step 3

Exam Tip

(A,B) के स्थानों के (4) जोड़े और (2) क्रम हैं, फिर शेष (5!) तरीके हैं। निश्चित दूरी में स्थान-जोड़े गिनें।

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कितनी (5)-अंकीय संख्याएं (0,1,2,3,4,5,6) से बिना पुनरावृत्ति बनेंगी जिनमें पहला अंक (6) हो या अंतिम अंक (6) हो?

How many (5)-digit numbers can be formed from (0,1,2,3,4,5,6) without repetition if the first digit is (6) or the last digit is (6)?

Explanation opens after your attempt
Correct Answer

A. (600)

Step 1

Concept

If first is (6), count \(^{6}P_{4}\); if last is (6), subtract leading-zero cases from choices; then subtract the intersection. Use inclusion-exclusion for 'or'.

Step 2

Why this answer is correct

The correct answer is A. (600). If first is (6), count \(^{6}P_{4}\); if last is (6), subtract leading-zero cases from choices; then subtract the intersection. Use inclusion-exclusion for 'or'.

Step 3

Exam Tip

पहला (6) होने पर \(^{6}P_{4}\), अंतिम (6) होने पर अग्र शून्य घटाकर \(5\cdot ^{5}P_{3}\), और दोनों स्थिति का प्रतिच्छेद घटता है। 'या' में समावेशन-बहिष्करण लगाएं।

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(6) अलग-अलग अंगूठियों को (4) अलग-अलग उंगलियों में पहनाने के कितने तरीके हैं यदि हर उंगली में अधिकतम एक अंगूठी हो?

In how many ways can (6) distinct rings be worn on (4) distinct fingers if each finger can have at most one ring?

Explanation opens after your attempt
Correct Answer

A. (360)

Step 1

Concept

For (4) fingers, choose and assign rings in \(^{6}P_{4}\) ways. Placing objects in distinct positions is an ordered selection.

Step 2

Why this answer is correct

The correct answer is A. (360). For (4) fingers, choose and assign rings in \(^{6}P_{4}\) ways. Placing objects in distinct positions is an ordered selection.

Step 3

Exam Tip

(4) उंगलियों के लिए (6) अंगूठियों में से क्रम सहित चयन \(^{6}P_{4}\) है। अलग स्थानों पर वस्तु रखना क्रम सहित चयन होता है।

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शब्द (EXAMINATION) में अक्षरों की कुल भिन्न व्यवस्थाओं की संख्या कितनी है?

What is the total number of distinct arrangements of the letters in (EXAMINATION)?

Explanation opens after your attempt
Correct Answer

A. (9979200)

Step 1

Concept

There are (11) letters with (I,N,A) each repeated twice, so the count is (11!/(2!2!2!)). In long words, note repeated letters separately.

Step 2

Why this answer is correct

The correct answer is A. (9979200). There are (11) letters with (I,N,A) each repeated twice, so the count is (11!/(2!2!2!)). In long words, note repeated letters separately.

Step 3

Exam Tip

(11) अक्षरों में (I,N,A) दो-दो बार हैं, इसलिए संख्या (11!/(2!2!2!)) है। बड़े शब्दों में दोहराए अक्षर अलग से नोट करें।

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(5) पुरुष और (3) महिलाएं गोल मेज पर कितने तरीकों से बैठ सकते हैं यदि सभी महिलाएं साथ बैठें?

In how many ways can (5) men and (3) women sit around a circular table if all women sit together?

Explanation opens after your attempt
Correct Answer

A. (720)

Step 1

Concept

Treat the three women as one block, so (6) units have (5!) circular arrangements and the women have (3!) internal ways. In circular block problems, count units carefully.

Step 2

Why this answer is correct

The correct answer is A. (720). Treat the three women as one block, so (6) units have (5!) circular arrangements and the women have (3!) internal ways. In circular block problems, count units carefully.

Step 3

Exam Tip

तीन महिलाओं को एक खंड मानकर (6) इकाइयों की गोल व्यवस्था (5!) है और महिलाओं के भीतर (3!) तरीके हैं। गोल खंड विधि में इकाइयों की संख्या ध्यान से लें।

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अंकों (2,3,4,5,6,7,8) से बिना पुनरावृत्ति कितनी (4)-अंकीय संख्याएं बनेंगी जो (4) से विभाज्य हों?

How many (4)-digit numbers divisible by (4) can be formed from digits (2,3,4,5,6,7,8) without repetition?

Explanation opens after your attempt
Correct Answer

A. (90)

Step 1

Concept

Divisibility by (4) depends on the last two digits; there are (15) valid ordered ending pairs and the first two places fill in \(^{5}P_{2}\) ways. For divisibility by (4), check the last two digits.

Step 2

Why this answer is correct

The correct answer is A. (90). Divisibility by (4) depends on the last two digits; there are (15) valid ordered ending pairs and the first two places fill in \(^{5}P_{2}\) ways. For divisibility by (4), check the last two digits.

Step 3

Exam Tip

अंतिम दो अंकों से (4) से विभाज्यता तय होती है; मान्य क्रमित अंतिम जोड़ों की संख्या (15) है और पहले दो स्थान \(^{5}P_{2}\) तरीकों से भरते हैं। (4) से विभाज्यता में अंतिम दो अंक जांचें।

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(6) अलग-अलग चित्रों को एक दीवार पर पंक्ति में कितने तरीकों से लगाया जा सकता है यदि दो विशेष चित्र बीच के दो स्थानों पर हों?

In how many ways can (6) distinct pictures be arranged in a row on a wall if two specified pictures occupy the two middle positions?

Explanation opens after your attempt
Correct Answer

B. (48)

Step 1

Concept

The two specified pictures occupy the two middle positions in (2!) ways, and the remaining (4) pictures in (4!) ways. Fill restricted positions first.

Step 2

Why this answer is correct

The correct answer is B. (48). The two specified pictures occupy the two middle positions in (2!) ways, and the remaining (4) pictures in (4!) ways. Fill restricted positions first.

Step 3

Exam Tip

दो विशेष चित्र बीच के (2) स्थानों पर (2!) तरीकों से और बाकी (4) चित्र (4!) तरीकों से लगेंगे। प्रतिबंधित स्थान पहले भरें।

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(5) अलग-अलग अक्षरों में से (3) अक्षरों के कितने क्रम बनेंगे यदि पुनरावृत्ति अनुमति नहीं है?

How many ordered arrangements of (3) letters can be formed from (5) distinct letters if repetition is not allowed?

Explanation opens after your attempt
Correct Answer

C. (60)

Step 1

Concept

This is an ordered selection of (3) from (5), so it is \(^{5}P_{3}\). When there is no repetition and order matters, use permutation.

Step 2

Why this answer is correct

The correct answer is C. (60). This is an ordered selection of (3) from (5), so it is \(^{5}P_{3}\). When there is no repetition and order matters, use permutation.

Step 3

Exam Tip

यह (5) में से (3) का क्रम सहित चयन है, इसलिए \(^{5}P_{3}\) होगा। बिना पुनरावृत्ति और क्रम हो तो क्रमचय लगाएं।

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कितनी व्यवस्थाओं में (9) अलग-अलग वस्तुओं में से (4) को चुना और क्रम में रखा जा सकता है?

In how many arrangements can (4) objects be chosen and ordered from (9) distinct objects?

Explanation opens after your attempt
Correct Answer

B. (3024)

Step 1

Concept

Both selection and order are involved, so the count is \(^{9}P_{4}\). When order matters, do not use combination.

Step 2

Why this answer is correct

The correct answer is B. (3024). Both selection and order are involved, so the count is \(^{9}P_{4}\). When order matters, do not use combination.

Step 3

Exam Tip

चयन और क्रम दोनों हैं, इसलिए \(^{9}P_{4}\) होगा। केवल चयन नहीं, क्रम भी हो तो संयोजन नहीं लगाना चाहिए।

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शब्द (PENCIL) की व्यवस्थाओं में (P) पहले और (L) अंतिम न हो, ऐसी कितनी व्यवस्थाएं हैं?

In arrangements of the word (PENCIL), how many have (P) first and (L) not last?

Explanation opens after your attempt
Correct Answer

A. (96)

Step 1

Concept

With (P) fixed first, subtract the (4!) arrangements where (L) is last from the remaining (5!). In mixed conditions, apply fixed conditions first.

Step 2

Why this answer is correct

The correct answer is A. (96). With (P) fixed first, subtract the (4!) arrangements where (L) is last from the remaining (5!). In mixed conditions, apply fixed conditions first.

Step 3

Exam Tip

(P) पहले निश्चित है; शेष (5!) में से (L) अंतिम होने वाली (4!) व्यवस्थाएं घटाएं। मिश्रित शर्तों में पहले निश्चित शर्त लगाएं।

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(7) विद्यार्थियों को एक पंक्ति में कितने तरीकों से बैठाया जा सकता है यदि दो विशेष विद्यार्थी पंक्ति के सिरों पर न बैठें?

In how many ways can (7) students sit in a row if two specified students do not sit at the ends?

Explanation opens after your attempt
Correct Answer

D. (3600)

Step 1

Concept

First fill the two ends with ordinary students in \(^{5}P_{2}\) ways, then arrange the remaining (5) students in the middle (5) places. For end restrictions, fill the ends first.

Step 2

Why this answer is correct

The correct answer is D. (3600). First fill the two ends with ordinary students in \(^{5}P_{2}\) ways, then arrange the remaining (5) students in the middle (5) places. For end restrictions, fill the ends first.

Step 3

Exam Tip

पहले (5) सामान्य विद्यार्थियों में से सिरों पर \(^{5}P_{2}\) तरीके और बीच के (5) स्थानों पर बाकी (5!) तरीके हैं। सिरों से जुड़ी शर्तों में पहले सिरों को भरना आसान है।

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शब्द (RANDOM) की व्यवस्थाओं में स्वर कभी साथ न आएं, ऐसी कितनी व्यवस्थाएं हैं?

In arrangements of the word (RANDOM), how many have no two vowels adjacent?

Explanation opens after your attempt
Correct Answer

C. (480)

Step 1

Concept

Arrange the four consonants in (4!) ways and place the two vowels in \(^{5}P_{2}\) ways in the gaps. Use the gap method for non-adjacent vowels.

Step 2

Why this answer is correct

The correct answer is C. (480). Arrange the four consonants in (4!) ways and place the two vowels in \(^{5}P_{2}\) ways in the gaps. Use the gap method for non-adjacent vowels.

Step 3

Exam Tip

चार व्यंजन (4!) तरीकों से बैठते हैं और (5) अंतरालों में (2) स्वरों को \(^{5}P_{2}\) तरीकों से रखा जाता है। न-साथ स्वर में अंतराल विधि लगाएं।

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(1,2,3,4,5,6,7,8,9) से बिना पुनरावृत्ति कितनी (5)-अंकीय संख्याएं बनेंगी जिनमें अंक (1) और (2) दोनों शामिल हों?

How many (5)-digit numbers can be formed from (1,2,3,4,5,6,7,8,9) without repetition if both digits (1) and (2) are included?

Explanation opens after your attempt
Correct Answer

A. (2520)

Step 1

Concept

Include (1,2), choose (3) more digits from the remaining (7), then arrange the (5) digits in (5!) ways. For compulsory objects, include them first and choose the rest.

Step 2

Why this answer is correct

The correct answer is A. (2520). Include (1,2), choose (3) more digits from the remaining (7), then arrange the (5) digits in (5!) ways. For compulsory objects, include them first and choose the rest.

Step 3

Exam Tip

(1,2) के साथ शेष (7) अंकों में से (3) चुनकर (5!) तरीकों से व्यवस्थित करें। अनिवार्य वस्तुओं को पहले शामिल मानकर बाकी चुनें।

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(6) अलग-अलग लोगों को (6) कुर्सियों पर बैठाने की कितनी व्यवस्थाएं होंगी यदि एक विशेष व्यक्ति अपनी निश्चित कुर्सी पर न बैठे?

How many arrangements are there for seating (6) distinct people on (6) chairs if one specified person does not sit on his fixed chair?

Explanation opens after your attempt
Correct Answer

A. (600)

Step 1

Concept

Subtract the (5!) arrangements where that person sits on the fixed chair from the total (6!). With one forbidden seat, subtract the invalid cases from total.

Step 2

Why this answer is correct

The correct answer is A. (600). Subtract the (5!) arrangements where that person sits on the fixed chair from the total (6!). With one forbidden seat, subtract the invalid cases from total.

Step 3

Exam Tip

कुल (6!) में से उस व्यक्ति के निश्चित कुर्सी पर बैठने की (5!) व्यवस्थाएं घटती हैं। एक निषिद्ध स्थान हो तो कुल से अनुकूल-नहीं घटाएं।

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