Permutation and Combination Problems and Solutions

• 15. 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 :

  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
  

• 16. Evaluate \begin{aligned}
  \frac{30!}{28!}
  \end{aligned}

  1. 970
  2. 870
  3. 770
  4. 670

  Answer :

  Option B

  Explanation:

  \begin{aligned}
  = \frac{30!}{28!} \\
  = \frac{30 * 29 * 28!}{28!} \\
  = 30 * 29 = 870
  \end{aligned}

• 17. How many words can be formed from the letters of the word "AFTER", so that the vowels never comes together.

  1. 48
  2. 52
  3. 72
  4. 120

  Answer :

  Option C

  Explanation:

  We need to find the ways that vowels NEVER come together.
  Vowels are A, E
  Let the word be FTR(AE) having 4 words.
  Total ways = 4! = 24
  Vowels can have total ways 2! = 2
  Number of ways having vowel together = 48
  Total number of words using all letter = 5! = 120
  
  Number of words having vowels never together = 120-48
  = 72

• 18. From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done

  1. 456
  2. 556
  3. 656
  4. 756

  Answer :

  Option D

  Explanation:

  From a group of 7 men and 6 women, five persons are to be selected with at least 3 men.
  So we can have
  (5 men) or (4 men and 1 woman) or (3 men and 2 woman)
  
  \begin{aligned}
  (^{5}{C}_{5}) + (^{5}{C}_{4} * ^{6}{C}_{1}) + \\
  + (^{5}{C}_{3} * ^{6}{C}_{2}) \\
  
  = \left[\dfrac{7 \times 6 }{2 \times 1}\right] + \left[\left( \dfrac{7 \times 6 \times 5}{3 \times 2 \times 1} \right) \times 6 \right] + \\ \left[\left( \dfrac{7 \times 6 \times 5}{3 \times 2 \times 1} \right) \times \left( \dfrac{6 \times 5}{2 \times 1} \right) \right] \\
  = 21 + 210 + 525 = 756
  \end{aligned}