Why is each interior entry in Pascal’s triangle the sum of the two entries above it?

In combinatorics, Pascal’s triangle is a geometric arrangement of binomial coefficients. The rule that each interior entry is the sum of the two entries directly above it is a direct consequence of the algebraic properties of the binomial theorem.

Specifically, the coefficients in the expansion of $(a+b{)}^n$ are generated by the relationship between successive powers. When you multiply $(a+b{)}^n$ by $(a+b)$ to obtain $(a+b{)}^{n+1}$, the coefficient of each term in the new expansion is formed by adding the coefficients of the corresponding terms from the previous expansion.

In terms of combinations, this is expressed by the identity:
$${}_n{C}_k{+}_n{C}_{k-1}{=}_{n+1}{C}_k$$
This formula confirms that the number of ways to choose $k$ objects from a set of $n+1$ is equal to the sum of the ways to choose $k$ objects from $n$ (excluding one specific object) and the ways to choose $k-1$ objects from $n$ (including that specific object). This additive property allows the triangle to be constructed row by row, starting with 1s on the outer edges.123