Intuitive explanation of binomial coefficient formula Regarding the formula for binomial coefficients:
$$\binom{n}{k}=\frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}$$
the professor described the formula as first choosing the $k$ objects from a group of $n$, where order matters, and then dividing by $k!$ to adjust for overcounting.
I understand the reasoning behind the numerator but don't understand why dividing by $k!$ is what's needed to adjust for overcounting. Can someone please help me understand how one arrives at $k!$? Thanks.

Suppose you had 5 books on a shelf and wanted to pick 3. If order mattered (permutations), there would be $5 \cdot 4 \cdot 3$ ways of doing so. If order did not matter (combinations), you would discount the extra permutations. That is, "ABC" would be the single representative of all of its $3!$ permutations. The total number ways of selecting the books is $\frac{5\cdot4\cdot3}{3!}$. For the general case, replace the 5 with $n$ and 3 with $k$.
