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(D) The word $ASSIGNMENT$ contains $10$ letters: $A, S, S, I, G, N, M, E, N, T$. The distinct letters are $\{A, S, I, G, N, M, E, T\}$ ($8$ distinct l

Vedclass · Accepted Answer

$2190$

Vedclass · Answer

(D) The word $ASSIGNMENT$ contains $10$ letters: $A, S, S, I, G, N, M, E, N, T$. The distinct letters are $\{A, S, I, G, N, M, E, T\}$ ($8$ distinct letters) and the repeated letters are $S$ (twice) and $N$ (twice).Case $1$: All $4$ letters are distinct.Number of ways to choose $4$ letters from $8$ is $\binom{8}{4} = 70$. Number of arrangements is $70 	imes 4! = 70 	imes 24 = 1680$.Case $2$: $2$ letters are same and $2$ are distinct.Subcase $i$: The pair is $S$ or $N$ ($2$ choices). Choose $2$ more from the remaining $7$ letters: $\binom{7}{2} = 21$. Arrangements: $2 	imes 21 	imes \frac{4!}{2!} = 42 	imes 12 = 504$.Case $3$: $2$ pairs of same letters.The pairs are $S$ and $N$ ($1$ choice). Arrangements: $\frac{4!}{2!2!} = 6$.Total number of words = $1680 + 504 + 6 = 2190$.

If four letters are chosen from the letters of the word $ASSIGNMENT$ and are arranged in all possible ways to form $4$-letter words (with or without meaning),then the total number of such words that can be formed is:

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