Permutation and Combination Questions Answers

• 8. In how many words can be formed by using all letters of the word BHOPAL

  1. 420
  2. 520
  3. 620
  4. 720

  Answer And Explanation

  Answer: Option D

  Explanation:

  Required number
  \begin{aligned}
  = 6! \\
  = 6*5*4*3*2*1 \\
  = 720
  \end{aligned}
  

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• 9. In how many way the letter of the word "APPLE" can be arranged

  1. 20
  2. 40
  3. 60
  4. 80

  Answer And Explanation

  Answer: Option C

  Explanation:

  Friends the main point to note in this question is letter "P" is written twice in the word.
  Easy way to solve this type of permutation question is as,
  
  So word APPLE contains 1A, 2P, 1L and 1E
  Required number =
  \begin{aligned}
  = \frac{5!}{1!*2!*1!*1!} \\
  = \frac{5*4*3*2!}{2!} \\
  = 60
  \end{aligned}
  

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• 10. In how many ways can the letters of the CHEATER be arranged

  1. 20160
  2. 2520
  3. 360
  4. 80

  Answer And Explanation

  Answer: Option B

  Explanation:

  As we can see the letter "E" is twice in given word, so Required Number
  \begin{aligned}
  = \frac{7!}{2!} \\
  = \frac{7*6*5*4*3*2!}{2!} \\
  = 2520
  \end{aligned}

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• 11. In how many way the letter of the word "RUMOUR" can be arranged

  1. 2520
  2. 480
  3. 360
  4. 180

  Answer And Explanation

  Answer: Option D

  Explanation:

  In above word, there are 2 "R" and 2 "U"
  So Required number will be
  
  \begin{aligned}
  = \frac{6!}{2!*2!} \\
  = \frac{6*5*4*3*2*1}{4} \\
  = 180
  \end{aligned}

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• 12. How many words can be formed from the letters of the word "SIGNATURE" so that vowels always come together.

  1. 17280
  2. 4320
  3. 720
  4. 80

  Answer And Explanation

  Answer: Option A

  Explanation:

  word SIGNATURE contains total 9 letters.
  There are four vowels in this word, I, A, U and E
  
  Make it as, SGNTR(IAUE), consider all vowels as 1 letter for now
  So total letter are 6.
  6 letters can be arranged in 6! ways = 720 ways
  Vowels can be arranged in themselves in 4! ways = 24 ways
  
  Required number of ways = 720*24 = 17280
  

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• 13. In how many ways can the letters of the word "CORPORATION" be arranged so that vowels always come together.

  1. 5760
  2. 50400
  3. 2880
  4. None of above

  Answer And Explanation

  Answer: Option B

  Explanation:

  Vowels in the word "CORPORATION" are O,O,A,I,O
  Lets make it as CRPRTN(OOAIO)
  
  This has 7 lettes, where R is twice so value = 7!/2!
  = 2520
  
  Vowel O is 3 times, so vowels can be arranged = 5!/3!
  
  = 20
  
  Total number of words = 2520 * 20 = 50400

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• 14. In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there

  1. 109
  2. 128
  3. 138
  4. 209

  Answer And Explanation

  Answer: Option D

  Explanation:

  In a group of 6 boys and 4 girls, four children are to be selected such that
  at least one boy should be there.
  So we can have
  
  (four boys) or (three boys and one girl) or (two boys and two girls) or (one boy and three gils)
  
  This combination question can be solved as
  
  \begin{aligned}
  (^{6}{C}_{4}) + (^{6}{C}_{3} * ^{4}{C}_{1}) + \\
  + (^{6}{C}_{2} * ^{4}{C}_{2}) + (^{6}{C}_{1} * ^{4}{C}_{3}) \\
  
  = \left[\dfrac{6 \times 5 }{2 \times 1}\right] + \left[\left(\dfrac{6 \times 5 \times 4 }{3 \times 2 \times 1}\right) \times 4\right] + \\\left[\left(\dfrac{6 \times 5 }{2 \times 1}\right)\left(\dfrac{4 \times 3 }{2 \times 1}\right)\right] + \left[6 \times 4 \right] \\
  = 15 + 80 + 90 + 24\\
  = 209
  \end{aligned}

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