Permutations & Combinations Questions and Answers

111.

In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?

Answer: C

MATHEMATICS has eleven letters, with M, A and T each appearing twice. Its vowels are A, E, A, I and its consonants M, T, H, M, T, C, S.

Treating the four vowels as one block gives eight units, among which M and T each repeat:

\[ \frac{8!}{2!\,2!} = \frac{40320}{4} = 10{,}080 \]

Within the block the four vowels include two A's:

\[ \frac{4!}{2!} = 12 \]

\[ 10{,}080 \times 12 = 120{,}960 \]

Three separate repetitions have to be divided out - the M's and T's among the units, and the A's inside the block.

112.

In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?

Answer: B

The word 'OPTICAL' contains 7 different letters.

When the vowels OIA are always together, they can be supposed to form one letter.

Then, we have to arrange the letters PTCL (OIA).

Now, 5 letters can be arranged in 5! = 120 ways.

The vowels (OIA) can be arranged among themselves in 3! = 6 ways.

Required number of ways = (120 x 6) = 720.

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