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

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 $2 x^2-2 x+2 y^2-2 y=x^2+2 x y+y^2-x-y$.
Rearranging, we have $x^2-2 x y+y^2=x+y \Rightarrow(x-y)^2=x+y \Rightarrow|x-y|=x+y$
Since $x+y \leq 1991, x-y \leq \sqrt{1991}$. Because $(\sqrt{1991})=44$, we get $x-y<44$.
We get the two inequalities
$x+y \leq(44)^2=1936 $
$x-y \leq 44$
Adding both together and dividing by two yields $x \leq 990$