Number of ways to split 8 people into groups of 2

Question

In how many ways can we split 8 people who are married into 4 (unlabeled) groups of two (each group consists of one man and one woman) such that no group is a married couple? So a solution is a set $\{(M_1,W_1),(M_2,W_2),(M_3,W_3),(M_4,W_4)\}$. Hint: for $i=1,2,3,4$, let $A_i$ be the set of groupings $\{(M_1,W_1),(M_2,W_2),(M_3,W_3),(M_4,W_4)\}$ such that married couple $i$ is one of the groups.

Answer 1

Let $A_i=\{\text{set of groupings such that married couple }i\text{, for $i=1,2,3,4$ is one of the groups}\}$ and $X=\{\text{# of ways to split 4 married couples in groups of 2 with one man and one woman}\}$. We know that $|X|=4!$ because if we first fix the 4 men, then there are 4! ways of seating the 4 women. For each $A_i$, there are $\left(\begin{array}{c} 4 \\ 1 \end{array}\right)$ ways to fix one married couple and then 3! ways to seat the remaining 3 women. Then the intersection of any two $A_i$'s would be $\left(\begin{array}{c} 4 \\ 2 \end{array}\right)$ ways to fix two married couples with 2! ways to seat the remaining 2 women. Similarly, for the intersection of three $A_i$'s, we get $\left(\begin{array}{c} 4 \\ 3 \end{array}\right)1!$ and the intersection of all 4 $A_i$'s would be $\left(\begin{array}{c} 4 \\ 4 \end{array}\right)0!$. By PIE, we have \begin{align*} |X\backslash(A_1\cup A_2\cup A_3\cup A_4)|=&4!-3!\left(\begin{array}{c} 4 \\ 1 \end{array}\right)+2!\left(\begin{array}{c} 4 \\ 2 \end{array}\right)-1!\left(\begin{array}{c} 4 \\ 3 \end{array}\right)+0!\left(\begin{array}{c} 4 \\ 4 \end{array}\right)\\ =&24-24+12-4+1\\ =&9. \end{align*}

Hence there are 9 ways we can split 8 people who are married into 4 (unlabeled) groups of two (each group consists of one man and one woman) such that no group is a married couple.