2008 AMC 10A Problem 23

Question

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?

20

40

60

160

320

Answer: B

Concepts:subsets combinations multiplication principle

Difficulty rating: 1770

Small hint:

First choose the two elements that belong to both subsets

Big hint:

Each of the remaining elements goes to exactly one subset; then correct for the unordered pair

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.