Given A ⊂ ℤ, |A| = 13, |B| = |C| = 3, and B, C ⊂ A, what is the probability that |C ∖ B| = 0?

Question

Given A ⊂ ℤ, |A| = 13, |B| = |C| = 3, and B, C ⊂ A, what is the probability that |C ∖ B| = 0?

Answer: _ _ _ _ _ _ _

Answer 1

A contains 13 integers. B and C each contain 3 integers that are contained in A, with possible overlap. There are 13C3 ways to choose 3 of the elements of A to create B, and 13C3 ways to do the same for C. Therefore there are 13C3*13C3 total choices for B and C, given A. The question is asking what is the probability that these two sets are actually the same. Well, there are 13C3 ways to choose those 3 elements that will belong to the (identical) sets B and C. So the total is

13C3 1 3!10! 6

----------- = ----------- = ----------- = ------------- = 1/286, or around 0.35%.

(13C3)^2 13C3 13! 13*12*11