The number of words formed from the letters o | vedclass

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(A) The word $RAJASTHAN$ contains $9$ letters: $R, A, J, A, S, T, H, A, N$.Consonants are: $R, J, S, T, H, N$ ($6$ letters).Vowels are: $A, A, A$ ($3$

Vedclass · Accepted Answer

$6! 	imes ^7C_3$

Vedclass · Answer

(A) The word $RAJASTHAN$ contains $9$ letters: $R, A, J, A, S, T, H, A, N$.Consonants are: $R, J, S, T, H, N$ ($6$ letters).Vowels are: $A, A, A$ ($3$ letters).First,arrange the $6$ consonants in $6!$ ways.To ensure vowels are alternate,we place them in the gaps created by the consonants.There are $7$ possible gaps (including ends) for $6$ consonants: $\_ C \_ C \_ C \_ C \_ C \_ C \_$.We need to choose $3$ gaps out of $7$ for the $3$ vowels,which can be done in $^7C_3$ ways.Since all $3$ vowels are identical $(A)$,there is only $1$ way to arrange them in the chosen gaps.Total number of words $= 6! 	imes ^7C_3$.

The number of words formed from the letters of the word $RAJASTHAN$ by taking all the letters at a time in which vowels are alternate is:

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