Question 13
Call an $2n$-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 $2n$-digit number is special. (We allow leading zeros in $2n$-digit numbers).
a. The sequence $p_{n}$ converges to a constant $c$. Find $c$.
b. Let $q_{n} = p_{n} - c$. There exists a unique positive constant $r$ such that $\frac{q_{n}}{r^{n}}$ converges to a constant $d$. Find $r$ and $d$.
a. The sequence $p_{n}$ converges to a constant $c$. Find $c$.
b. Let $q_{n} = p_{n} - c$. There exists a unique positive constant $r$ such that $\frac{q_{n}}{r^{n}}$ converges to a constant $d$. Find $r$ and $d$.