[Siegmund]

Here is a simpler way to count.

First, select which matched pairs from the 10 available ones to draw.

Then, select which of the remaining pairs you will draw one glove from.

Then, for each single, select whether it is a left or right glove.

0 pairs: 10C6 * 2^6 = 13,440.

1 pair: 10 * 9C4 * 2^4 = 20,160.

2 pairs: 10C2 * 8C2 * 2^2 = 5,040.

3 pairs: 10C3 = 120.

Your only error was that 3C3=1, not 3, on your "3 pairs" line.

EV: 30600 / 38760 ~ 0.7894 pairs.

The "Expected number of pairs picked = 10 * the chance that any given pair is picked" argument is correct.

The chance of picking one particular pair, however, is 18C4 / 20C6 = 3060 / 38760, rather than the number quoted by the other responders.