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

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(D) The word $BARRACK$ consists of $7$ letters: $A, A, R, R, B, C, K$. The distinct letters are ${A, R, B, C, K}$.We need to form $4$-letter words. Th

Vedclass · Accepted Answer

$270$

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(D) The word $BARRACK$ consists of $7$ letters: $A, A, R, R, B, C, K$. The distinct letters are ${A, R, B, C, K}$.We need to form $4$-letter words. The cases are:$(i)$ All $4$ letters are distinct: We choose $4$ letters from ${A, R, B, C, K}$ in $^5C_4 = 5$ ways. Each set can be arranged in $4! = 24$ ways. Total $= 5 	imes 24 = 120$.(ii) $2$ pairs of identical letters: The pairs are ${A, A}$ and ${R, R}$. We choose both pairs in $^2C_2 = 1$ way. The number of arrangements is $\frac{4!}{2!2!} = 6$.(iii) $2$ identical letters and $2$ distinct letters: We choose $1$ pair from ${A, A}$ or ${R, R}$ in $^2C_1 = 2$ ways. We then choose $2$ distinct letters from the remaining $4$ letters in $^4C_2 = 6$ ways. The number of arrangements for each selection is $\frac{4!}{2!} = 12$. Total $= 2 	imes 6 	imes 12 = 144$.Total number of $4$-letter words $= 120 + 6 + 144 = 270$.

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

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