Suppose I have a set of $n$ boys and each subset of $1\leqslant k \leqslant n$ boys knows at least $k$ girls then this means that the set of boys $B$ satisfies the marriage condition. I am doing an exercise from a Graph Theory book and it says we can show by induction on $n$ that the number of possible marriage arrangements is $\geqslant t!$ if every individual boy knows at least $t \leqslant n$ girls or there are $\geqslant t!/(t-n)!$ possible marriages if each individual boy knows at least $t>n$ girls.
I am struggling to see how the induction hypothesis can be implemented to prove the $n+1$ case. When you swap from the $n$ to $n+1$ case the conditions change and I can't see the link between them.
Would anyone be able to point me in the right direction?