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 4

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