2008 AMC 10A Problem 23

Problem 23 of 25HarderCounting & Probability

Two subsets of the set

S = \{a, b, c, d, e\}

are to be chosen so that their union is

S

and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?

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Solution

Choose the two common elements in

\binom{5}{2} = 10

ways. Each of the remaining

3

elements must lie in exactly one subset, giving

2^3 = 8

assignments, for

80

ordered pairs. Since the order of the two subsets does not matter, divide by

2

to get

\dfrac{80}{2} = 40.

Thus, the correct answer is B.

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