2000 AIME II Problem 3

Question

A deck of forty cards consists of four

1

’s, four

2

’s,

\ldots,

and four

10

’s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let

\frac{m}{n}

be the probability that two randomly selected cards also form a pair, where

m

and

n

are relatively prime positive integers. Find

m + n.

Answer

After the pair is removed,

38

cards remain: nine numbers with four cards each and one number with only two

Count the favorable pairs as

9\binom{4}{2} + \binom{2}{2}

out of

\binom{38}{2}

equally likely draws

After the matching pair is removed,

38

cards remain: nine numbers with four cards each and one number with only two cards. The number of ways to draw a pair is

9\binom{4}{2} + \binom{2}{2} = 54 + 1 = 55,

out of

\binom{38}{2} = 703

equally likely draws.

Since

703 = 19 \cdot 37

shares no factor with

55 = 5 \cdot 11,

the probability

\frac{55}{703}

is in lowest terms, and

m + n = 55 + 703 = 758.

Problem 3 in other years