Pascal's Formula

Each subsequent row of Pascal's Triangle is obtained by adding the two entries diagonally above. This follows immediately from the Binomial Coefficient identity

$$\begin{eqnarray*} {n\choose r} &\equiv& {n!\over (n-r)!r!} = {(n-1)!n\over (n-r)... ...)!\over (n-r)!(r-1)!}\\ &=& {n-1\choose r} + {n-1\choose r-1}. \end{eqnarray*}$$

See also Binomial Coefficient, Pascal's Triangle