Is there any way to simplify this into a combinatorial formula?
$$\frac{t!(n-t)!}{n!}$$
It is the reciprocal of a binomial coefficient, assuming $0 \le t \le n$:
$$ \frac{t!(n-t)!}{n!} = \binom{n}{t}^{-1} $$
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It is the reciprocal of a binomial coefficient, assuming $0 \le t \le n$:
$$ \frac{t!(n-t)!}{n!} = \binom{n}{t}^{-1} $$