The definition of $\binom{n}{k}$ is the number of $k$ -subsets out from an $n$ -set. Using this combinatorial meaning the proof is straightforward.

Let $x$ a distinct element from the $n$ -set. As previously stated, $\binom{n}{k}$ counts the number of subsets with $k$ elements, chosen from the set with $n$ elements. Now, some of these subsets will contain $x$ and some others don't.

The number of $k$ -subsets not containing $x$ is $\binom{n-1}{k}$ , since we need to choose $k$ elements from the $n-1$ elements different from $x$ .

The number of $k$ -subsets containing $x$ is $\binom{n-1}{k-1}$ , because if it is given that $x$ is in the subset, we only need to choose the remaining $k-1$ elements from the $n-1$ elements that are different from $x$ .

Thus$$ \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}.$$