Question

The number of ways the letters of the word MATHEMATICS be permuted so that the vowels never comes together

• A. $\frac{11!}{2!2!2!}$
• B. $\frac{8!4!}{2!2!2!}$
• C. $\frac{7!5!}{2!2!2!}$
• D. None of these

Answer 1

Answer: (D)

None of these

MATHEMATICS we are having 4 vowels they are A,A,E,I which can be arranged in 8 gaps _M_T_H_M_T_C_S_ in $\frac{{}^8{\text{P}}_4}{2!}$ ways as A is repeating twice.
And the rest 7 letters can be arranged in $\frac{7!}{2!2!}$ as M is repeating twice and T is also repeating.
$\therefore$ the number of ways are $\frac{{}^8{\text{P}}_4}{2!}\times \frac{7}{2!2!}$
=$\frac{7!8!}{2!2!4!2!}$
Hence option D is correct