The number of ways five alphabets can be chos | 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: $\{M, A, T, H, E, I, C, S\}$. We

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$179$

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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: $\{M, A, T, H, E, I, C, S\}$. We need to choose $5$ letters. The cases are: $(1)$ All $5$ letters are distinct: We choose $5$ letters from $8$ distinct letters: $^8C_5 = \frac{8 	imes 7 	imes 6}{3 	imes 2 	imes 1} = 56$. $(2)$ $2$ letters are same (one pair) and $3$ are distinct: There are $3$ pairs $(M, M)$,$(A, A)$,$(T, T)$. We choose $1$ pair in $^3C_1$ ways. From the remaining $7$ distinct letters,we choose $3$ letters in $^7C_3$ ways. Number of ways = $^3C_1 	imes ^7C_3 = 3 	imes 35 = 105$. $(3)$ $2$ pairs of same letters and $1$ distinct letter: We choose $2$ pairs from $3$ available pairs in $^3C_2$ ways. From the remaining $6$ distinct letters,we choose $1$ letter in $^6C_1$ ways. Number of ways = $^3C_2 	imes ^6C_1 = 3 	imes 6 = 18$. Total number of ways = $56 + 105 + 18 = 179$.

The number of ways five alphabets can be chosen from the alphabets of the word $MATHEMATICS$,where the chosen alphabets are not necessarily distinct,is equal to :

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