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 want to plant three identical red flowers in a row. In how many ways can you do it? First you plant all three red flowers in a row. Suppose you swap the first flower with the second one. Did anything change? You still have three red flowers in a row. Try swapping the first flower with the third one. Do you see any difference? I can't. They are still the same three red flowers planted in a row. Basically, no matter how your permute these flowers, you have only one outcome, namely a row of three identical red flowers. 
So, there are $3!$ ways to plant these flowers in a row, but since every row looks exactly like the other row, we divide out the redundant ones. There are $3!$ redundant rows. Then $\frac {3!}{3!} = 1$ way to plant three identical red flowers. 
