The number of all possible combinations of 4 | vedclass

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Question

(A) The word $ACCOMMODATION$ contains $13$ letters: $A-2, C-2, O-3, M-2, D-1, T-1, I-1, N-1$. The distinct letters are ${A, C, O, M, D, T, I, N}$,tota

Vedclass · Accepted Answer

$167$

Vedclass · Answer

(A) The word $ACCOMMODATION$ contains $13$ letters: $A-2, C-2, O-3, M-2, D-1, T-1, I-1, N-1$. The distinct letters are ${A, C, O, M, D, T, I, N}$,total $8$ distinct letters.We need to select $4$ letters. The following cases arise:$1$. $3$ alike and $1$ different: We choose $1$ letter from ${O}$ (the only letter appearing $3$ times) and $1$ letter from the remaining $7$ distinct letters. Number of ways $= {}^{1}C_{1} 	imes {}^{7}C_{1} = 1 	imes 7 = 7$.$2$. $2$ alike and $2$ alike: We choose $2$ letters from the $4$ letters that appear twice ${A, C, O, M}$. Number of ways $= {}^{4}C_{2} = 6$.$3$. $2$ alike and $2$ different: We choose $1$ letter from the $4$ letters that appear twice,and $2$ letters from the remaining $7$ distinct letters. Number of ways $= {}^{4}C_{1} 	imes {}^{7}C_{2} = 4 	imes 21 = 84$.$4$. All $4$ different: We choose $4$ letters from the $8$ distinct letters. Number of ways $= {}^{8}C_{4} = 70$.Total number of ways $= 7 + 6 + 84 + 70 = 167$.

The number of all possible combinations of $4$ letters which are taken from the letters of the word $ACCOMMODATION$ is

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