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

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(D) The word $BARRACK$ contains $7$ letters: $A, A, R, R, B, C, K$. The distinct letters are ${A, R, B, C, K}$.Case $1$: All $4$ letters are distinct.

Vedclass · Accepted Answer

$270$

Vedclass · Answer

(D) The word $BARRACK$ contains $7$ letters: $A, A, R, R, B, C, K$. The distinct letters are ${A, R, B, C, K}$.Case $1$: All $4$ letters are distinct.We choose $4$ letters from ${A, R, B, C, K}$ in $^5C_4 = 5$ ways. These $4$ letters can be arranged in $4! = 24$ ways.Total words $= 5 	imes 24 = 120$.Case $2$: Two letters are the same (a pair) and two are distinct.Subcase $2a$: The pair is $R, R$. We choose $2$ other distinct letters from ${A, B, C, K}$ in $^4C_2 = 6$ ways. The number of arrangements is $\frac{4!}{2!} = 12$ ways.Total words $= 6 	imes 12 = 72$.Subcase $2b$: The pair is $A, A$. We choose $2$ other distinct letters from ${R, B, C, K}$ in $^4C_2 = 6$ ways. The number of arrangements is $\frac{4!}{2!} = 12$ ways.Total words $= 6 	imes 12 = 72$.Case $3$: Two pairs of letters ($A, A$ and $R, R$).We choose $2$ pairs from ${A, A, R, R}$ in $^2C_2 = 1$ way. The number of arrangements is $\frac{4!}{2!2!} = 6$ ways.Total words $= 1 	imes 6 = 6$.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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