All the letters of the word 'EAMCET' | vedclass

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Question

(C) The word '$EAMCET$' consists of $6$ letters: $E, A, M, C, E, T$.
The vowels are $E, A, E$ (where $E$ repeats twice) and the consonants ar

Vedclass · Accepted Answer

$72$

Vedclass · Answer

(C) The word '$EAMCET$' consists of $6$ letters: $E, A, M, C, E, T$.
The vowels are $E, A, E$ (where $E$ repeats twice) and the consonants are $M, C, T$.
First,we arrange the $3$ consonants $(M, C, T)$ in $3! = 6$ ways.
These $3$ consonants create $4$ gaps (represented by underscores): $ M C T $.
We need to place the $3$ vowels $(E, A, E)$ in these $4$ gaps such that no two vowels are adjacent.
The number of ways to choose $3$ gaps out of $4$ is $^4C3 = 4$.
Since there are two identical vowels $(E)$,the number of ways to arrange $E, A, E$ in the chosen $3$ gaps is $\frac{3!}{2!} = 3$.
Total arrangements = (Ways to arrange consonants) $	imes$ (Ways to choose gaps) $	imes$ (Ways to arrange vowels in gaps)
Total arrangements = $6 	imes 4 	imes 3 = 72$.

All the letters of the word '$EAMCET$' are arranged in all possible ways. The number of such arrangements in which two vowels are not adjacent to each other is

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