Question

All the letters of the word ${}^{\prime }EAMCO{T}^{\prime }$ are arranged in different possible ways. Find the number of arrangement in which no two vowels are adjacent to each other.

Answer 1

To find no. of ways of arranging 'EAMCOT' in which no two vowels are adjacent to each other, first arrange consonants.

Vowels: E, A, D

Consonants: M, C, T

_M_C_T_$\to$ can be arranged in $3!$ ways

Now, fill the blanks with vowels, it can be done in ${}^4{P}_3$ ways.

So, total no. of ways$=3!\times {}^4{P}_3$

$\Rightarrow 6\times \frac{4!}{1!}$

$\Rightarrow 144$ ways.