1991 AIME Problems/Problem 13

Problem

A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?

Solution

Let and denote the number of red and blue socks, respectively. Also, let .

The probability that when two socks are drawn without replacement, both are red or both are blue is given by

$P=\frac{r(r-1)}{(r+b)(r+b-1)}+\frac{b(b-1)}{(r+b)(r+b-1)}=\frac{r(r-1)+b(b-1)}{t(t-1)}=\frac{1}{2} <math>

== See also == {{AIME box|year=1991|num-b=12|num-a=14}}$ (Error compiling LaTeX. Unknown error_msg)