Consider the students as variables $a_i\in\{0; 1\}$ depending on the color of their shirts. One coloring will be represented as the $n-$tuple $(a_1, a_2,\dots, a_n)$. Notice that if we can achieve a set $A$ from $B$, the we can achieve $B$ from $A$, just changing the same clubs. Given a set $S$, we can change the clubs in $2^k$ different ways, given that changing a club twice does nothing. Now, we know that, from any set $S$ we can achieve a set $T$ with at least $n-1$ equal elements, thus for any set $S$, there is a set $T$ with at least $n-1$ equal elements from which we can achieve $S$. A simple counting will tell us there are $2n+2$ such sets $T$. Since we can operate $2^k$ different ways in each $T$, the amout of $T's$ times the amount of operations will give us at least the total amount of sets: $(2n+2)\cdot2^k\geq 2^n\Rightarrow n\geq 2^{n-k-1}-1. _{\blacksquare}$