I'm not quite sure if I'm understanding solution to following problem: "Determine the number of ways for n couples to stand in a line so that no one stands beside his or her partner."
The general approach seems to be to:
Allow |U| to be the total numbers of arrangements possible for n couples (or 2n people); then, |U| = (2n)!
Let $X_i$ = Set of arrangements in which couple i is standing together; then $X_1\cup X_2 \cup X_3 \ldots \cup X_n$ is the set of arrangements where one or more couples are standing together.
I'm fine with the set up until this point, but I'm not sure how the this portion is derived:
$|X_{i_1} \cap X_{i_2} \cup \ldots X_n| = 2^k(2n-2k+k)$
I believe $2^k$ comes from the number of permutations per couple - e.g., each couple can stand in two ways. However, I'm not sure how the $(2n-2k+k)$ was derived.
If someone could help to explain, that'd be great - thanks!
For $k$ specific couples we want $2^k (2n-2k+k)!$. Your explanation of the $2^k$ is correct. For the rest, put the two members of each of the $k$ couples in a bag. That leaves $2n-2k$ people and $k$ bags, a total of $2n-2k+k$ objects, which can be arranged in $(2n-2k+k)!$ ways.