In how many different ways can the letters of the word CORPORATION be arranged so that the vowels may occupy only the odd positions?

Solution (By Examveda Team)

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