Discussion

In how many ways can the letters of the word ENCYCLOPAEDIA be arranged such that vowels only occupy the even positions?

Answer: A

ENCYCLOPAEDIA has thirteen letters: vowels E, O, A, E, A, I (six, with E and A each twice) and consonants N, C, Y, C, L, P, D (seven, with C twice).

Positions 2, 4, 6, 8, 10 and 12 are the six even ones - exactly enough for the vowels.

\[ \text{vowels} = \frac{6!}{2!\,2!} = \frac{720}{4} = 180 \]

\[ \text{consonants in the 7 odd positions} = \frac{7!}{2!} = 2520 \]

\[ 180 \times 2520 = 453{,}600 \]

The counts of vowels and even positions match exactly, which is what makes the arrangement possible at all.

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