The number of ways in which four letters of t | 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 $8$ distinct letters: $\{M, A, T, H, E, I, C, S\}$.Ca

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$2454$

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(D) The word '$MATHEMATICS$' contains $11$ letters: $M, M, A, A, T, T, H, E, I, C, S$. There are $8$ distinct letters: $\{M, A, T, H, E, I, C, S\}$.Case $I$: $2$ alike of one kind and $2$ alike of another kind.Number of ways to choose $2$ pairs from $3$ pairs $(M, A, T)$ is $^3C_2 = 3$.Number of arrangements = $3 	imes \frac{4!}{2!2!} = 3 	imes 6 = 18$.Case $II$: $2$ alike of one kind and $2$ different.Number of ways to choose $1$ pair from $3$ pairs is $^3C_1 = 3$. Number of ways to choose $2$ different letters from the remaining $7$ distinct letters is $^7C_2 = 21$.Number of arrangements = $3 	imes 21 	imes \frac{4!}{2!} = 63 	imes 12 = 756$.Case $III$: All $4$ letters are different.Number of ways to choose $4$ different letters from $8$ distinct letters is $^8C_4 = 70$.Number of arrangements = $70 	imes 4! = 70 	imes 24 = 1680$.Total number of arrangements = $18 + 756 + 1680 = 2454$.

The number of ways in which four letters of the word '$MATHEMATICS$' can be arranged is given by

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