Mary has $6$ identical basil plants, and three different window sills she can put them on. How many ways are there for Mary to put the plants on the window sills?
I want to count the number of ways to distribute $6$ identical objects among $3$ distinct bins. This is a combinatorics problem, and I recall that there is a formula for this situation: the number of ways to distribute $n$ identical objects among $k$ distinct bins is given by the binomial coefficient $\binom{n+k-1}{k-1}$. In this case, $n=6$ and $k=3$, so the formula gives me $\binom{6+3-1}{3-1}=\binom{8}{2}$. To calculate this, I need to know the definition of the binomial coefficient: $\binom{n}{k}=\frac{n!}{k!(n-k)!}$, where $n!$ is the factorial of $n$, which means the product of all positive integers up to $n$. Plugging in the values, I get $\binom{8}{2}=\frac{8!}{2!(8-2)!}=\frac{8\times 7\times 6!}{2\times 6!}$. Simplifying, I see that the $6!$ terms cancel out, and I get $\binom{8}{2}=\frac{8\times 7}{2}=\frac{56}{2}=28$. So there are $28$ ways for Mary to put the plants on the window sills. 