In how many different ways can the letters of the word CORPORATION be arranged so that the vowels may occupy only the odd positions?
A
810
B
1440
C
2880
D
50400
E
None of these
Correct Answer: 50400
Keeping the vowels (OOAIO) together as one letter we have CRPRTN (OOAIO).This has 7 letters, out of which we have 2R, 1C, 1P, 1T and 1N.
Number of ways of arranging three letters
$$\eqalign{ & = \frac{{7!}}{{2!}} \cr & = \frac{{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}} \cr & = 2520 \cr} $$
Now, (OOAIO) has 5 letters, out of which we have 3O, 1A and 1I.
Number of ways of arranging these letters
$$\eqalign{ & = \frac{{5!}}{{3!}} \cr & = \frac{{5 \times 4 \times 3 \times 2 \times 1}}{{3 \times 2 \times 1}} \cr & = 20 \cr} $$
∴ Required number of ways = (2520 × 20) = 50400
