The number of words which can be formed from | vedclass

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Question

(A) The word $MAXIMUM$ contains $7$ letters: $M, A, X, I, M, U, M$. Consonants are $M, X, M, M$ ($4$ letters) and vowels are $A, I, U$ ($3$ letters).

Vedclass · Accepted Answer

$4!$

Vedclass · Answer

(A) The word $MAXIMUM$ contains $7$ letters: $M, A, X, I, M, U, M$. Consonants are $M, X, M, M$ ($4$ letters) and vowels are $A, I, U$ ($3$ letters). To ensure no two consonants occur together,we use the gap method. First,arrange the $3$ vowels $(A, I, U)$ in $3!$ ways. These $3$ vowels create $4$ possible gaps (including ends): $\_ V \_ V \_ V \_$. We need to place the $4$ consonants $(M, X, M, M)$ into these $4$ gaps. The number of ways to arrange $4$ consonants in $4$ gaps is $\frac{4!}{3!}$ (since $M$ repeats $3$ times). Total number of words = (Arrangement of vowels) $	imes$ (Arrangement of consonants in gaps) $= 3! 	imes \frac{4!}{3!} = 4! = 24$.

The number of words which can be formed from the letters of the word $MAXIMUM$,if two consonants cannot occur together,is

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