The number of different words that can be for | vedclass

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Question

(D) The word $MORADABAD$ consists of $9$ letters: $A, A, A, D, D, M, O, R, B$. There are $3$ $A$'s,$2$ $D$'s,and $4$ distinct letters $(M, O, R, B)$.

Vedclass · Accepted Answer

$626$

Vedclass · Answer

(D) The word $MORADABAD$ consists of $9$ letters: $A, A, A, D, D, M, O, R, B$. There are $3$ $A$'s,$2$ $D$'s,and $4$ distinct letters $(M, O, R, B)$. We need to form $4$-letter words.$(i)$ All $4$ letters are different: We choose $4$ letters from the $6$ distinct types $(A, D, M, O, R, B)$. Number of ways = $^6P_4 = 6 	imes 5 	imes 4 	imes 3 = 360$.$(ii)$ $2$ letters are alike and $2$ are different: We choose $1$ pair from $2$ types ($A$ or $D$) and $2$ distinct letters from the remaining $5$ types. Number of ways = $^2C_1 	imes ^5C_2 	imes \frac{4!}{2!} = 2 	imes 10 	imes 12 = 240$.$(iii)$ $3$ letters are alike and $1$ is different: We choose $1$ triplet $(A)$ and $1$ distinct letter from the remaining $5$ types. Number of ways = $^1C_1 	imes ^5C_1 	imes \frac{4!}{3!} = 1 	imes 5 	imes 4 = 20$.$(iv)$ $2$ letters are alike of one type and $2$ are alike of another type: We choose $2$ pairs from $2$ types ($A$ and $D$). Number of ways = $^2C_2 	imes \frac{4!}{2!2!} = 1 	imes 6 = 6$.Total number of words = $360 + 240 + 20 + 6 = 626$.

The number of different words that can be formed out of the letters of the word $MORADABAD$ taken four at a time is

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