In how many ways can 4 letters be selected an | vedclass

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(D) The word $MATHEMATICS$ contains $11$ letters: $M, M, A, A, T, T, H, E, I, C, S$. There are $3$ pairs of identical letters $(M, A, T)$ and $5$ dist

Vedclass · Accepted Answer

$2454$

Vedclass · Answer

(D) The word $MATHEMATICS$ contains $11$ letters: $M, M, A, A, T, T, H, E, I, C, S$. There are $3$ pairs of identical letters $(M, A, T)$ and $5$ distinct letters $(H, E, I, C, S)$.Case $I$: Two letters of one kind and two of another kind.Number of ways $= ^3C_2 	imes \frac{4!}{2!2!} = 3 	imes 6 = 18$.Case $II$: Two letters of one kind and two distinct letters.Number of ways $= ^3C_1 	imes ^7C_2 	imes \frac{4!}{2!} = 3 	imes 21 	imes 12 = 756$.Case $III$: All four letters are distinct.Number of ways $= ^8C_4 	imes 4! = 70 	imes 24 = 1680$.Total number of arrangements $= 18 + 756 + 1680 = 2454$.

In how many ways can $4$ letters be selected and arranged from the word $MATHEMATICS$?

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