Combinations of Pairs of Gloves..halp

[yellowjack]

There are 10 pairs of gloves (20 gloves), and 6 are selected at random. Let X be the number of pairs of gloves. Find the EV of X.

I'm having trouble getting all the possible combinations to add up to 20_C_6 = 38,760 . Can someone help me by looking at my numbers?

I distributed the possible combinations by splitting the 20 gloves up into left and right, 2 groups of 10 (i.e. the gloves are labelled L1, L2,..,L10, R1, R2,...,R10)

The 2nd combination for 0 pairs is "choose 5 from the 10 Left gloves, then choose 5 of the remaining Right gloves whose pair has not been chosen". For example, if L1, L2,...,L5 were chosen, one of R6,R7,...,R10 would be chosen with it.

0 pairs
(10_C_6 * 4_C_0) + (10_C_5 * 5_C_1) + (10_C_4 * 6_C_2) + (10_C_3 * 7_C_3) + (10_C_2 * 8_C_4) + (10_C_1 * 9_C_5) + (4_C_0 * 10_C_6) = 13,440

1 pair
(10_C_5 * 5_C_1 * 5_C_0) + (10_C_4 * 4_C_1* 6_C_1) + (10_C_3 * 3_C_1 * 7_C_2) + (10_C_2 * 2_C_1 * 8_C_3) + (10_C_1 * 1_C_1 * 9_C_4) = 20,160

2 pairs
(10_C_4 * 4_C_2 * 6_C_0) + (10_C_3 * 3_C_2 * 7_C_1) + (10_C_2 * 2_C_2 * 8_C_2) = 5,040

3 pairs
10_C_3 * 3_C_3 = 360

20_C_6 = 38,760
The total of combinations from the above is 39,000; 240 too many. Can you spot the flaw in the combinations? Also if you have a different counting scheme that may/may not be more efficient than mine, please post it. Thanks.