The number of total possible combinations can be calculated using the Binomial Coefficient. The Binomial Coefficient is so common that it receives it's own notation: $\binom{n}k$ or "$n$ choose $kquot;. If its helpful you can say "from $n$ choose $kquot; to clarify that $n$ is the number of objects in the set and $k$ is the number chosen. Before we consider the formula for the binomial coefficient, let's motivate our understanding with an example. How many ways can you select a committee of 3 people from a group of 10 people? We know we can calculate the number of ways to order all 10 people using the factorial ($10!$). But we only want to select 3 people. We don't care about the ways to order the final 7 people, so we can divide by the ways to order 7 people ($7!$). We also don't care about order, so we could further divide by the ways to order 3 people ($3!$). To review, to calculate the ways to choose 3 people ($k$) from 10 people ($n$), we've calculated the ways to order 10 people ($n!$) and divided by the ways to order the remaining 7 people ($(n-k)!$) and then the ways to order the 3 chosen people ($k!$) to get the number of ways to select 3 people in any order from 10 people. The formula of the binomial coefficient is: $\frac{n!}{(n-k)! (k!)}$ ## Symmetry of the binomial coefficient The binomial coefficient is symmetric. $\displaystyle \binom{n}{k}$ equals $\displaystyle \binom{n}{k-1}$. Consider selecting a team on the playground. Because all of the kids will either be selected or not selected, picking who is on the team is the same thing as picking who is not on the team.