1991 AIME Problems/Problem 13

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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 $\displaystyle \frac{1}{2}$ 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?


Let $r$ and $b$ denote the number of red and blue socks, respectively. Also, let $t=r+b$.

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

\begin{eqnarray} 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} \nonumber \end{eqnarray}

See also

1991 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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