Exercise 1.39

Question

An organization with 2n people consists of n married couples. A committee of size k is
selected, with all possibilities equally likely. Find the probability that there are exactly
j married couples within the committee.

fifthist x · Accepted Answer

There are a total of $\binom{2N}{K}$ possible committees of $K$ people. 
There are $\binom{N}{j}$ ways to select $j$ couples for the committee. 
$K-2j$ people need to be selected from the remaining $N-j$ couples such 
that only one person is selected from a couple. First, we select $K-2j$ 
couples from the remaining $N-j$ couples. Then, for each of the selected 
couples, there are $2$ choices for committee membership.

$$\frac{\binom{N}{j}\binom{N-j}{K-2j}2^{K-2j}}{\binom{2N}{K}}$$

Exercise 1.39

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Exercise 1.39

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