Question #40306

Let $A, B,$ and $C$ be three subsets of a universal set $U$ where $|U| = 100$. It is given that $|A| + |B| + |C| = 120$, $|A \cap B| + |B \cap C| + |C \cap A| = 37$, and $|A \cap B \cap C| = 5$. Let $n_k$ (for $k=1,2,3$) denote the number of elements of $U$ belonging to exactly $k$ of these sets, and $n_0$ denote the number of elements belonging to none of the sets. If $K$ is the coefficient of $x_0^{n_0} x_1^{n_1} x_2^{n_2} x_3^{n_3}$ in the expansion of $(x_0 + x_1 + x_2 + x_3)^{100}$, find the exponent of $5$ in the prime factorization of $K$.