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

Answer: Option D

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
$$\begin{array}{rl}& =\frac{7!}{2!}\\ & =\frac{7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 1}\\ & =2520\end{array}$$
Now, (OOAIO) has 5 letters, out of which we have 3O, 1A and 1I.
Number of ways of arranging these letters
$$\begin{array}{rl}& =\frac{5!}{3!}\\ & =\frac{5\times 4\times 3\times 2\times 1}{3\times 2\times 1}\\ & =20\end{array}$$
∴ Required number of ways = (2520 × 20) = 50400