# proof of Pascal’s rule

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

 $\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}.$
Title proof of Pascal’s rule ProofOfPascalsRule 2013-03-22 15:03:11 2013-03-22 15:03:11 drini (3) drini (3) 5 drini (3) Proof msc 05A19