There are 4 balls of different colors and 4 b | vedclass

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Question

(A) We need to select $2$ balls out of $4$ and place them in the boxes corresponding to the other $2$ balls such that no ball goes into its own box.Nu

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(A) We need to select $2$ balls out of $4$ and place them in the boxes corresponding to the other $2$ balls such that no ball goes into its own box.Number of ways to select $2$ balls out of $4$ is $^4C_2 = \frac{4 	imes 3}{2 	imes 1} = 6$.For the $2$ selected balls,the number of ways to place them such that neither goes into its own box is given by the derangement $D_2 = 2! 	imes (1 - 1 + \frac{1}{2!}) = 1$.Thus,the total number of ways is $6 	imes 1 = 6$.

There are $4$ balls of different colors and $4$ boxes of the same colors as the balls. In how many ways can $2$ balls be placed in the boxes such that they do not match their respective colors?

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