In how many ways can the letters of the word | vedclass

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(A) The word '$MATHEMATICS$' consists of $11$ letters: $M, A, T, H, E, M, A, T, I, C, S$.The vowels are $A, A, E, I$ (total $4$ vowels).The consonants

Vedclass · Accepted Answer

$120960$

Vedclass · Answer

(A) The word '$MATHEMATICS$' consists of $11$ letters: $M, A, T, H, E, M, A, T, I, C, S$.The vowels are $A, A, E, I$ (total $4$ vowels).The consonants are $M, T, H, M, T, C, S$ (total $7$ consonants).Treating the $4$ vowels as a single unit,we have $7$ consonants + $1$ unit of vowels = $8$ entities to arrange.Among these $8$ entities,the letter $M$ repeats $2$ times and $T$ repeats $2$ times.The number of ways to arrange these $8$ entities is $\frac{8!}{2! 	imes 2!} = \frac{40320}{4} = 10080$.Within the vowel unit,the $4$ vowels $(A, A, E, I)$ can be arranged in $\frac{4!}{2!}$ ways (since $A$ repeats $2$ times).$\frac{4!}{2!} = \frac{24}{2} = 12$.Total number of arrangements = $10080 	imes 12 = 120960$.

In how many ways can the letters of the word '$MATHEMATICS$' be arranged,when all vowels come together?

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