Question

In how many ways can the letters of the word "CORPORATION" be arranged so that vowels always occupy even places?

• A. 120
• B. 720
• C. 2700
• D. 7200

Hint:

When vowels are restricted to certain positions, count arrangements for vowels and consonants separately, then multiply.

Answer 1

The Correct Option is D

Step 1: The word "CORPORATION" has 11 letters.
Step 2: Identify the vowels and consonants:
Vowels: O, O, A, I, O (5 vowels, with O repeated 3 times).
Consonants: C, R, P, R, T, N (6 consonants, with R repeated twice).
Step 3: In an arrangement where vowels must be in even positions, note that the positions are numbered: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.
Even positions = 2, 4, 6, 8, 10 → exactly 5 even positions.
Step 4: Place the vowels in these even positions. Since there are exactly 5 vowels, all even positions will be occupied by vowels.
The number of arrangements of vowels: \[ \frac5!3! = \frac1206 = 20 \] (the 3! accounts for repetition of O three times).
Step 5: The remaining 6 positions (odd positions) will be occupied by the 6 consonants.
Number of arrangements of consonants: \[ \frac6!2! = \frac7202 = 360 \] (the 2! accounts for repetition of R twice).
Step 6: Total arrangements = Arrangements of vowels × Arrangements of consonants: \[ 20 \times 360 = 7200 \] Thus, the answer is $\mathbf7200$.