1.2.1 problem 17

$\left(\frac{2n}{n}\right)$ counts the number of ways to sample $n$ objects from a set of $2n$. Instead of sampling from the whole set, we can break the set into two sets of size $n$ each. Then, we have to sample $n$ objects in total from both sets.

We can sample all $n$ objects from the first set, or $1$ object from the first set and $n-1$ objects from the second set, or $2$ objects from the first set and $n-2$ objects from the second set and so on.

There are $\left(\frac{n}{n}\right)$ ways to sample all $n$ objects from the first set, $\left(\frac{n}{1}\right)\left(\frac{n}{n-1}\right)$ ways to sample $1$ object from the first set and $n-1$ objects from the second set, $\left(\frac{n}{2}\right)\left(\frac{n}{n-2}\right)$ ways to sample $2$ objects from the first set and $n-2$ objects from the second set. The pattern is clear