5 nonsimilar pairs of socks are in a closet 4 socks are chos

5 non-similar pairs of socks are in a closet. 4 socks are chosen at random. what is the probability that there will be among the 4 socks chosen

(a) No complete pair

(b) Exactly 1 complete pair

(c) Exactly 2 complete pairs

Answer (a) 0.381 (b) 0.571 (c) 0.048

Solution

There are 10 socks in the closet, and so there are
10C4 = 10! / (6! 4!) = 10 * 9 * 8 * 7 / (4 * 3 * 2 * 1)
= 10 * 3 * 7 = 210 possible combinations of 4 socks that can be chosen randomly.

a

If there is no complete pair, then the combination consists of
1 sock (out of 2 possible) from 4 different pairs,
and there are 5 possibilities for the pair that is not represented at all.
So there are
2^4 * 5 = 80 such combinations possible, and the probability of such a combination is
80/210 = 8/21

b

If there is exactly one complete pair, it could be one of 5 possible pairs.
The remaining two socks represent 2 of the remaining 4 pairs, which can happen in
4C2 = 4! / (2! 2!) = 4 * 3 / 2 = 6 ways,
and again there are 2 possible socks which can represent each of those pairs.
So there are
5 * 6 * 2 * 2 = 120 such combinations, and the probability is
120/210 = 4/7

c

If the 4 socks constitute 2 complete pairs, there are
5C2 = 5! / (3! 2!) = 10 possible combinations of pairs, so the probability is
10/210 = 1/21