All pairs out of four

Question

This is a problem that's bugging me and I can't seem to unravel it.

A friend mentioned that they do a trip over Christmas week each year. They have a family of four and they rotate, each pair planning the vacation in secret that year and surprising the others the day before they leave. He said they're on a five year cycle.

On the way home, I did some figuring:

1) Dad & Mom 2) Dad & Son 3) Mom & Daughter 4) Dad & Daughter 5) Mom & Son

and then there's son and daughter, which would be six.

To get all possible combinations of four, you use factorial, so it would be 4! or 1 * 2 * 3 * 4 = 24 possible combinations. But I can't figure out for the life of me how to calculate all possible pairs out of a group of four. The answer appears to be six, but can't come up with a proof.

Answer 1

Since order doesn't matter, here, we just use the formula for combinations:

n!/(k!(n-k)!)

In this case, this is just four choose two. Or 4!/(2!(4-2)!), which equals 6.