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Question

(A) Total ways to distribute $5$ identical samples among $12$ houses is given by $\binom{12}{5} = \frac{12 	imes 11 	imes 10 	imes 9 	imes 8}{5

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$\frac{7}{99}$

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(A) Total ways to distribute $5$ identical samples among $12$ houses is given by $\binom{12}{5} = \frac{12 	imes 11 	imes 10 	imes 9 	imes 8}{5 	imes 4 	imes 3 	imes 2 	imes 1} = 792$. To find the number of ways such that no two consecutive houses get a sample,we use the gap method. Place the $7$ houses that do not receive a sample in a row. This creates $8$ possible gaps (including the ends) where the $5$ samples can be placed. The number of ways to choose $5$ gaps out of $8$ is $\binom{8}{5} = \binom{8}{3} = \frac{8 	imes 7 	imes 6}{3 	imes 2 	imes 1} = 56$. The probability is $\frac{56}{792} = \frac{7}{99}$.

$A$ company representative is distributing $5$ identical samples of a product among $12$ houses in a row such that each house gets at most one sample. The probability that no two consecutive houses get one sample is

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