Show that if a set of $n$ boys satisfies Hall’s marriage condition and each individual boy knows at least $t>n$ girls then the number of possible marriage arrangements between boys and girls is greater than or equal to $t!/(t-n)!$ or if each individual boy knows at least $t\leqslant n$ girls then the number of possible marriage arrangements is greater than or equal to $t!$
The question suggests proof by induction on $n$.
This is an exercise from Introduction to Graph Theory by Robin J Wilson.