The number of four-letter words that can be f | vedclass

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Question

(C) The word $BARRACK$ contains $7$ letters: $A, A, R, R, B, C, K$. Case $I$: All four letters are different. We choose $4$ letters from ${A, R, B, C,

Vedclass · Accepted Answer

$270$

Vedclass · Answer

(C) The word $BARRACK$ contains $7$ letters: $A, A, R, R, B, C, K$. Case $I$: All four letters are different. We choose $4$ letters from ${A, R, B, C, K}$. Number of ways $= {}^{5}C_{4} 	imes 4! = 5 	imes 24 = 120$. Case $II$: Two letters are the same $(R, R)$ and two are different. We choose $2$ letters from ${A, B, C, K}$. Number of ways $= {}^{4}C_{2} 	imes \frac{4!}{2!} = 6 	imes 12 = 72$. Case $III$: Two letters are the same $(A, A)$ and two are different. We choose $2$ letters from ${R, B, C, K}$. Number of ways $= {}^{4}C_{2} 	imes \frac{4!}{2!} = 6 	imes 12 = 72$. Case $IV$: Two letters are $A, A$ and two are $R, R$. Number of ways $= \frac{4!}{2!2!} = 6$. Total number of words $= 120 + 72 + 72 + 6 = 270$.

The number of four-letter words that can be formed using the letters of the word $BARRACK$ is:

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