A. यह \(^{7}C_4\times4!\) है/It is \(^{7}C_4\times4!\)
Step 1
Concept
First choose (4) letters and then arrange them in (4!) orders. In exams order is important in words.
Step 2
Why this answer is correct
The correct answer is A. यह \(^{7}C_4\times4!\) है / It is \(^{7}C_4\times4!\). First choose (4) letters and then arrange them in (4!) orders. In exams order is important in words.
Step 3
Exam Tip
पहले (4) letters चुनते हैं फिर उन्हें (4!) orders में रखते हैं। परीक्षा में words में order महत्वपूर्ण होता है।
There are (4) even positions among (9), and the (4) consonants occupy them in (4!) ways, while vowels fill the rest in (5!) ways. For position restrictions, first count the positions.
Step 2
Why this answer is correct
The correct answer is A. (2880). There are (4) even positions among (9), and the (4) consonants occupy them in (4!) ways, while vowels fill the rest in (5!) ways. For position restrictions, first count the positions.
Step 3
Exam Tip
(9) स्थानों में (4) सम स्थान हैं और (4) व्यंजन उन्हीं में (4!) तरीकों से आएंगे, स्वर (5!) तरीकों से बची जगहों पर आएंगे। स्थान-प्रतिबंध में पहले स्थानों की संख्या देखें।
Here (S) appears three times and (I) appears three times, so the count is (11!/(3!3!)). Divide by the factorial of each repeated group.
Step 2
Why this answer is correct
The correct answer is D. (1108800). Here (S) appears three times and (I) appears three times, so the count is (11!/(3!3!)). Divide by the factorial of each repeated group.
Step 3
Exam Tip
इसमें (S) तीन बार और (I) तीन बार है, इसलिए संख्या (11!/(3!3!)) है। दोहराव के हर समूह के फैक्टोरियल से भाग दें।
Here (T) appears three times and (O,N,I) appear twice each, so the count is (12!/(3!2!2!2!)). For long words, make a frequency table.
Step 2
Why this answer is correct
The correct answer is A. (9979200). Here (T) appears three times and (O,N,I) appear twice each, so the count is (12!/(3!2!2!2!)). For long words, make a frequency table.
Step 3
Exam Tip
इसमें (T) तीन बार और (O,N,I) दो-दो बार हैं, इसलिए संख्या (12!/(3!2!2!2!)) है। लंबे शब्द में आवृत्ति तालिका बनाएं।
Inside the vowel block there are (5!/(2!3!)) arrangements, and outside there are (6!/2!) arrangements. Handle the block and repetitions together.
Step 2
Why this answer is correct
The correct answer is A. (3600). Inside the vowel block there are (5!/(2!3!)) arrangements, and outside there are (6!/2!) arrangements. Handle the block and repetitions together.
Step 3
Exam Tip
स्वर-खंड के भीतर (5!/(2!3!)) और बाहर (6!/2!) व्यवस्थाएं हैं। खंड और दोहराव दोनों को एक साथ देखें।
Only the (3) vowel positions must be chosen, because both vowel and consonant orders are fixed. When orders are fixed, position selection gives the answer.
Step 2
Why this answer is correct
The correct answer is D. (56). Only the (3) vowel positions must be chosen, because both vowel and consonant orders are fixed. When orders are fixed, position selection gives the answer.
Step 3
Exam Tip
केवल स्वरों के (3) स्थान चुनने हैं, क्योंकि स्वर और व्यंजन दोनों के क्रम निश्चित हैं। क्रम निश्चित हो तो स्थान-चयन ही उत्तर देता है।
Add arrangements beginning with smaller available letters at each position and then add (1). Write the alphabetical order first when finding rank.
Step 2
Why this answer is correct
The correct answer is B. (18635). Add arrangements beginning with smaller available letters at each position and then add (1). Write the alphabetical order first when finding rank.
Step 3
Exam Tip
प्रत्येक स्थान पर छोटे उपलब्ध अक्षरों से बनने वाली व्यवस्थाएं जोड़ें और अंत में (1) जोड़ें। रैंक निकालते समय वर्णक्रम पहले लिखें।
There are (5) odd positions among (9); place the (3) vowels in \(^{5}C_3\cdot 3!/2!\) ways and consonants in (6!/2!) ways. Decide position types first.
Step 2
Why this answer is correct
The correct answer is B. (10800). There are (5) odd positions among (9); place the (3) vowels in \(^{5}C_3\cdot 3!/2!\) ways and consonants in (6!/2!) ways. Decide position types first.
Step 3
Exam Tip
(9) स्थानों में (5) विषम स्थान हैं, जिनमें (3) स्वर \(^{5}C_3\cdot 3!/2!\) तरीकों से बैठेंगे और व्यंजन (6!/2!) तरीकों से। स्थान-प्रकार पहले तय करें।
Arrange the (5) consonants in (5!) ways and place the vowels in (3) of the (6) gaps in \(^{6}P_3\) ways. If vowels are distinct, count both positions and order.
Step 2
Why this answer is correct
The correct answer is C. (14400). Arrange the (5) consonants in (5!) ways and place the vowels in (3) of the (6) gaps in \(^{6}P_3\) ways. If vowels are distinct, count both positions and order.
Step 3
Exam Tip
पहले (5) व्यंजनों को (5!) तरीकों से रखें और (6) अंतरालों में से (3) में स्वरों को \(^{6}P_3\) तरीकों से रखें। स्वर अलग हों तो स्थान और क्रम दोनों गिनें।
Choose the (5) vowel positions in \(^{13}C_5\) ways and arrange consonants in (8!/2!) ways. When vowel order is fixed, do not count their separate orders.
Step 2
Why this answer is correct
The correct answer is D. (25945920). Choose the (5) vowel positions in \(^{13}C_5\) ways and arrange consonants in (8!/2!) ways. When vowel order is fixed, do not count their separate orders.
Step 3
Exam Tip
स्वरों के (5) स्थान \(^{13}C_5\) तरीकों से चुनें और व्यंजनों को (8!/2!) तरीकों से रखें। स्वर-क्रम निश्चित होने पर उनके अलग क्रम नहीं गिने जाते।
Choose consonant positions in \(^{11}C_6\) ways and arrange the (5) vowels in (5!) ways. When relative order is fixed, only positions are chosen.
Step 2
Why this answer is correct
The correct answer is B. (55440). Choose consonant positions in \(^{11}C_6\) ways and arrange the (5) vowels in (5!) ways. When relative order is fixed, only positions are chosen.
Step 3
Exam Tip
व्यंजनों के स्थान \(^{11}C_6\) तरीकों से चुनें और शेष (5) स्वरों को (5!) तरीकों से रखें। सापेक्ष क्रम निश्चित हो तो केवल स्थान चुनने होते हैं।
Treat the vowels as one block, giving (8!/(2!2!)) outside and (4!/2!) inside. Do not treat repeated letters as distinct.
Step 2
Why this answer is correct
The correct answer is C. (120960). Treat the vowels as one block, giving (8!/(2!2!)) outside and (4!/2!) inside. Do not treat repeated letters as distinct.
Step 3
Exam Tip
स्वरों को एक खंड मानकर बाहर (8!/(2!2!)) और भीतर (4!/2!) तरीके मिलते हैं। दोहराए अक्षरों को हर बार अलग न मानें।
Arrange consonants in (7!/(2!2!)) ways, then choose (4) of the (8) gaps and arrange vowels in (4!/(2!2!)) ways. Use the gap method for non-adjacent vowels.
Step 2
Why this answer is correct
The correct answer is C. (529200). Arrange consonants in (7!/(2!2!)) ways, then choose (4) of the (8) gaps and arrange vowels in (4!/(2!2!)) ways. Use the gap method for non-adjacent vowels.
Step 3
Exam Tip
पहले व्यंजन (7!/(2!2!)) तरीकों से रखें, फिर (8) अंतरालों में (4) स्वर चुनकर (4!/(2!2!)) तरीकों से सजाएं। न-साथ स्वर में अंतराल विधि लगाएं।
It has (N) three times, (E) three times and (D) twice, so the count is (11!/(3!3!2!)). Write frequencies before applying the formula.
Step 2
Why this answer is correct
The correct answer is C. (554400). It has (N) three times, (E) three times and (D) twice, so the count is (11!/(3!3!2!)). Write frequencies before applying the formula.
Step 3
Exam Tip
इसमें (N) तीन बार, (E) तीन बार और (D) दो बार है, इसलिए संख्या (11!/(3!3!2!)) है। आवृत्ति लिखकर ही सूत्र लगाएं।
Add arrangements starting with smaller available letters at each position, then add (1). In rank problems, first arrange letters alphabetically.
Step 2
Why this answer is correct
The correct answer is D. (140). Add arrangements starting with smaller available letters at each position, then add (1). In rank problems, first arrange letters alphabetically.
Step 3
Exam Tip
हर स्थान पर छोटे उपलब्ध अक्षरों से शुरू होने वाली व्यवस्थाएं जोड़कर अंत में (1) जोड़ते हैं। रैंक में अक्षरों को पहले वर्णक्रम में रखें।
Inside the vowel block there are (5!/(2!2!)) arrangements, and outside there are (6!/2!) arrangements. Do not forget repeated letters when forming blocks.
Step 2
Why this answer is correct
The correct answer is D. (10800). Inside the vowel block there are (5!/(2!2!)) arrangements, and outside there are (6!/2!) arrangements. Do not forget repeated letters when forming blocks.
Step 3
Exam Tip
स्वर-खंड के भीतर (5!/(2!2!)) और बाहर (6!/2!) व्यवस्थाएं हैं। खंड बनाते समय दोहराए अक्षरों को न भूलें।
Here (E,T,I) are each repeated twice, so the count is (10!/(2!2!2!)). Divide by the factorial of each repeated letter.
Step 2
Why this answer is correct
The correct answer is B. (453600). Here (E,T,I) are each repeated twice, so the count is (10!/(2!2!2!)). Divide by the factorial of each repeated letter.
Step 3
Exam Tip
इसमें (E,T,I) दो-दो बार हैं, इसलिए संख्या (10!/(2!2!2!)) है। हर दोहराए अक्षर के फैक्टोरियल से भाग दें।
Inside the vowel block there are (4!/2!) arrangements, and outside there are (8!/2!) arrangements. Divide for repeated consonants and vowels.
Step 2
Why this answer is correct
The correct answer is C. (241920). Inside the vowel block there are (4!/2!) arrangements, and outside there are (8!/2!) arrangements. Divide for repeated consonants and vowels.
Step 3
Exam Tip
स्वर-खंड के भीतर (4!/2!) और बाहर (8!/2!) व्यवस्थाएं हैं। दोहराए व्यंजन और स्वर दोनों से भाग दें।
It has (I) four times, (S) four times and (P) twice, so the count is (11!/(4!4!2!)). For long words, make a frequency table.
Step 2
Why this answer is correct
The correct answer is A. (34650). It has (I) four times, (S) four times and (P) twice, so the count is (11!/(4!4!2!)). For long words, make a frequency table.
Step 3
Exam Tip
इसमें (I) चार बार, (S) चार बार और (P) दो बार है, इसलिए संख्या (11!/(4!4!2!)) है। लंबा शब्द हो तो आवृत्ति तालिका बनाएं।
In the total (8!) arrangements, the (4!) relative orders of consonants are equally likely, so the count is (8!/4!). For original-order conditions, divide by relative orders.
Step 2
Why this answer is correct
The correct answer is B. (1680). In the total (8!) arrangements, the (4!) relative orders of consonants are equally likely, so the count is (8!/4!). For original-order conditions, divide by relative orders.
Step 3
Exam Tip
कुल (8!) व्यवस्थाओं में (4) व्यंजनों के (4!) सापेक्ष क्रम बराबर हैं, इसलिए (8!/4!) है। मूल क्रम वाली शर्त में सापेक्ष क्रमों से भाग दें।
Inside the vowel block there are (4!/3!) arrangements, and outside there are (5!/2!) arrangements. Handle repetition and blocks together.
Step 2
Why this answer is correct
The correct answer is B. (240). Inside the vowel block there are (4!/3!) arrangements, and outside there are (5!/2!) arrangements. Handle repetition and blocks together.
Step 3
Exam Tip
स्वर-खंड के भीतर (4!/3!) और बाहर (5!/2!) व्यवस्थाएं होंगी। दोहराव और खंड दोनों को साथ संभालें।
Treat vowels (A,I) as one block, giving \(5!\cdot 2!\) arrangements. For together conditions, the block method is most useful.
Step 2
Why this answer is correct
The correct answer is C. (240). Treat vowels (A,I) as one block, giving \(5!\cdot 2!\) arrangements. For together conditions, the block method is most useful.
Step 3
Exam Tip
स्वरों (A,I) को एक खंड मानने पर \(5!\cdot 2!\) व्यवस्थाएं मिलती हैं। साथ वाली शर्त में खंड विधि सबसे उपयोगी है।
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।
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।
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।
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।
(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।
