Question

Consider the letters of the word 'MATHEMATICS'. Possible number of words in which no two vowels are together is

• A. $7{!}^8{C}_4\frac{4!}{2!}$
• B. ${\frac{7!}{2!}}^8{C}_4\frac{4!}{2!}$
• C. ${\frac{7!}{2!2!}}^8{C}_4\frac{4!}{2!}$
• D. ${\frac{7!}{2!2!2!}}^8{C}_4\frac{4!}{2!}$

Answer 1

The correct option is C ${\frac{7!}{2!2!}}^8{C}_4\frac{4!}{2!}$

Consonants can be placed in $\frac{7!}{2!2!}$ ways.
Then there are $8$ places and $4$ vowels.
Therefore, number of ways is $=\frac{7!}{2!2!}{.}^8{C}_4\frac{4!}{2!}$