Question 22
For every positive integer $n$, let $f(n)$ be the number of all positive integers with exactly $2n$ digits, each having exactly $n$ of the digits equal to $1$ and the other $n$ digits equal to $2$. Let $g(n)$ be the number of all positive integers with exactly $n$ digits, each of its digits can only be $1$, $2$, $3$ or $4$ and the number of $1$'s equals the number of $2$'s. Prove that $f(n) = g(n)$.