Solution
= Solution
Define the <binomial coefficient> $\binom nr$ as the number of $r$-element <subset>[subsets] of an $n$-element set, equivalently as $n!/[r!(n-r)!]$. Partitioning the $r$-element subsets of $\{1,\ldots,n+1\}$ according to whether they contain $n+1$ proves <Pascal's identity>
$$
\binom{n+1}r=\binom nr+\binom n{r-1}.
$$
Thus the <forward difference operator> satisfies
$$
\boxed{(\delta f_r)(n)=\binom{n+1}r-\binom nr=\binom n{r-1}=f_{r-1}(n).}
$$