What's the intuition behind this equality involving combinatorics? What is the intuition behind
$$
\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}
$$
? I can't grasp why picking a group of $k$ out of $n$ bijects to first picking a group of $k-1$ out of $n-1$ and then a group of $k$ out of $n-1$.
 A: You don't pick a group of $k-1$ out of $n-1$ and a group of $k$ out of $n-1$; this would correspond to multiplication. You should do one or the other.
The key is to pick one of your $n$ objects as being special. Then any collection of $k$ objects from the $n$ either consists of $k-1$ objects from the $n-1$ non-special objects, plus the special one, or it consists entirely of $k$ non-special objects. There are $\binom{n-1}{k}$ collections of $k$ non-special objects, and $\binom{n-1}{k-1}$ collections of $k$ objects including the special one.
A: We have a group of $n$ people, one of whom is John. We want to pick a committee of $k$ people. By definition this can be done in $\binom{n}{k}$ ways. 
There are $\binom{n-1}{k}$ committees of $k$ that don't include John, for we can choose any $k$ of the people other than John. And there are $\binom{n-1}{k-1}$ committees of $k$ that do include John, for we can choose any $k-1$ people to join John. 
Note that automatically a committee that doesn't include John is different from a committee that includes John. So we have divided the $\binom{n}{k}$ possible committees into two groups, one of which has $\binom{n-1}{k}$ elements, and the other of which has $\binom{n-1}{k-1}$ elements. 
