HMMT 2010 T10

Call an $2 n$-digit base-10 number special if we can split its digits into two sets of size $n$ such that the sum of the numbers in the two sets is the same. Let $p_{n}$ be the probability that a randomly-chosen $2 n$-digit number is special. (We allow leading zeros in $2 n$-digit numbers). (a) [20] The sequence $p_{n}$ converges to a constant $c$. Find $c$.