In how many ways can the letters of the word MATHEMATICS be arranged so that all the vowels always come together?
A
10080
B
120960
C
4989600
D
21160
E
None of these
Correct Answer: 120960
Keeping the vowels (AEIA) together, we have MTHMTCS (AEAI).Now, we have to arrange 8 letters, out of which we have 2M, 2T and the rest are all different.
Number of ways of arranging these letters
$$\eqalign{ & = \frac{{8!}}{{2!.2!}} \cr & = \frac{{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 2 \times 1}} \cr & = 10080 \cr} $$
Now, (AEAI) has 4 letters, out of which we have 2A, 1E and 1I.
Number of ways of arranging these letters
$$\eqalign{ & = \frac{{4!}}{{2!}} \cr & = \frac{{4 \times 3 \times 2 \times 1}}{2} \cr & = 12 \cr} $$
∴ Required number of ways = (10080 × 12) = 120960
