The number of 5-letter words that can be form | vedclass

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(C) The word $SARANAM$ contains $7$ letters: $A, A, A, S, R, N, M$. The distinct letters are $\{A, S, R, N, M\}$.We need to form $5$-letter words. The

Vedclass · Accepted Answer

$480$

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(C) The word $SARANAM$ contains $7$ letters: $A, A, A, S, R, N, M$. The distinct letters are $\{A, S, R, N, M\}$.We need to form $5$-letter words. The possible cases are:$(i)$ All $5$ letters are different (using $S, A, R, N, M$):Number of words $= 5! = 120$.(ii) $2$ letters are alike (specifically $A$) and $3$ are different:We choose $3$ letters from the remaining $4$ distinct letters $\{S, R, N, M\}$ in $^4C_3$ ways. The number of arrangements is $\frac{5!}{2!} = 60$.Total words $= ^4C_3 	imes 60 = 4 	imes 60 = 240$.(iii) $3$ letters are alike (specifically $A$) and $2$ are different:We choose $2$ letters from the remaining $4$ distinct letters $\{S, R, N, M\}$ in $^4C_2$ ways. The number of arrangements is $\frac{5!}{3!} = 20$.Total words $= ^4C_2 	imes 20 = 6 	imes 20 = 120$.Total number of words $= 120 + 240 + 120 = 480$.

The number of $5$-letter words that can be formed by using the letters of the word $SARANAM$ is

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