Question

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 1 2 that both are red or both are blue. Find the largest possible number of red socks in the drawer that is consistent with this data

Answer 1

Answer: 990

Let there be $x$ red socks and $y$ blue socks. Then $\frac{{}^x{C}_2+{}^y{C}_2}{{}^{x+y}{C}_2}=\frac{1}{2}$
Let there be $x$ red socks and $y$ blue socks. Then $\frac{C_2+C_2}{{}^{x+y}{C}_2}=\frac{1}{2}$
or $\frac{x(x-1)+y(y-1)}{(x+y)(x+y-1)}=\frac{1}{2}$
Multiplying both sides by $2(x+y)(x+y-1)$ and expanding,
we find that $2x^2-2x+2y^2-2y=x^2+2xy+y^2-x-y$.
Rearranging, we have $x^2-2xy+y^2=x+y\Rightarrow (x-y{)}^2=x+y\Rightarrow |x-y|=x+y$
Since $x+y\le 1991,x-y\le \sqrt{1991}$. Because $(\sqrt{1991})=44$, we get $x-y<44$.
We get the two inequalities
$x+y\le (44{)}^2=1936$
$x-y\le 44$
Adding both together and dividing by two yields $x\le 990$