From the $r$-class variations that can be formed from $n$ elements, how many of them include the first $s$ elements?
We note that we are dealing with permutations without repetition and $s<r$.

Permutations without repetition can be created by permuting combinations without repetition. Therefore, we first solve the problem in terms of combinations.

If we fix the first $s$ elements, we need to add $r-s$ more elements to them; we choose these from the remaining $n-s$ elements. The number of such selections is $\binom{n-s}{r-s}$, which is the number of combinations in which these first $s$ elements occur. Each of these combinations, where the number of elements is $r$, can be permuted in $r!$ different orders, and thus we obtain the permutations, the number of which is

$$
\binom{n-s}{r-s} r!=\frac{(n-s)!r!}{(r-s)!(n-r)!}
$$

Hoffmann Tibor (Szent István g. VI. o. Bp. XIV.)