Can you solve this without pen and paper?
Suppose $r \gt 0$ is a fixed positive integer.
Can you give a formula (in terms of $n$ and $r$) for
$$ \sum_{k=r}^{n} \left( \prod_{j=0}^{r} (k-j) \right)$$
i.e.
$$\sum_{k=r}^{n} k(k-1)(k-2)\dots(k-r)$$
[Solution]
Suppose $r \gt 0$ is a fixed positive integer.
Can you give a formula (in terms of $n$ and $r$) for
$$ \sum_{k=r}^{n} \left( \prod_{j=0}^{r} (k-j) \right)$$
i.e.
$$\sum_{k=r}^{n} k(k-1)(k-2)\dots(k-r)$$
[Solution]
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