Problem #DES-091124_2

Problem

From the examples above, we see that we often need to pick $k$ objects from $n$ objects where the order of the $k$ objects is ignored. The number of ways to pick them is notated with the special symbol $(\genfrac{}{}{0ex}{}{n}{k})$, pronounced “$n$ choose $k$". Following a similar line of reasoning as the examples, we can write down a general formula:

$$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n\times (n-1)\times \dots (n-k+1)}{k\times (k-1)\times \dots 1}=\frac{n!}{k!(n-k)!}.$$

$n!$ is a shorthand for $n\times (n-1)\times \cdots \times 1$, pronounced “$n$ factorial". This is another useful expression and allows us to write down many formulas succinctly.