A drawer contains 6 blue socks and 4 green socks. In how many ways can 3 socks be selected such that both blue socks and green socks are selected?

Answer 1

Since 3 socks total are being selected and we want socks of both colors to be selected, we are being asked to calculate the number of ways of getting 2 Blue socks and 1 Green Sock or 1 Blue sock and 2 Green socks. Converting to use of (obvious) symbols representing the sets involved, find

n(2 B out of 6 B & 1 G out of 4 G or 1B out of 6 B & 2 G out of 4 G)

= n(2 B out of 6 B) • n(1 G out of 4 G) + n(1 B out of 6 B) • n(2 G out of 4 G)

= C(6,2) • C(4,1) + C(6,1) • C(4,2)

= 15 • 4 + 6 • 6

= 60 + 36

= 96 // The "or" means we are dealing with the Union of events, so we Add and the "and" gives us the intersection where we Multiply. The C(n,k) = n! / [(n-k)! (k!)] is for the number of ways to do combinations (order of selections doesn't matter) and there is no replacement after selecting (cannot draw same again).