Discussion

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

Answer: D

CORPORATION has eleven letters - O three times, R twice - with vowels O, O, A, I, O and consonants C, R, P, R, T, N.

Treating the five vowels as one block gives seven units, among which R repeats:

\[ \frac{7!}{2!} = 2520 \]

Within the block, the five vowels include three O's:

\[ \frac{5!}{3!} = 20 \]

\[ 2520 \times 20 = 50{,}400 \]

Both repetitions have to be divided out - the R's among the units and the O's inside the block.

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