Let $S$ be a subset of $\{1,2,3,\dots,100 \}$ of size $20$.

Let $P_3(S)$ denote the subsets of $S$ with $3$ elements. Consider the sums of the elements in each of the $3$-element subsets in $P_3(S)$

The set $P_3(S)$ must contain $k$ subsets whose elements sum to the same number. What is the __smallest__ possible value of $k$?