In how many different ways can the letters of the word CORPORATION be arranged in such a way that the vowels always come together?
Correct Answer: Description for Correct answer:
There are 11 letters in the word 'CORPORATION' of which three are O's, two are R's and all others are distinct. There are 5 vowels viz, O, O, O, I, A.
Considering these 5 vowels as one letter we have 7 letters as (C. R. R, N. T, P and letter obtained by combining all vowels) out of which R occurs twice.
These 7 letters can be arranged in \( \Large \frac{7!}{5!} \) ways.
But the 5 vowels (O, O, O, I, A) can be put together in \( \Large \frac{5!}{3!} \) ways
Hence. the number of arrangements in which vowels are always together
= \( \Large \frac{7!}{2!} \times \frac{5!}{3!} \)
= \( \Large \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 5 \times 4 \times 3 \times 2}{2 \times 3 \times 2} \)
= 50400
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