The figure shows the first seven rows of the arrangement of numbers known as Pascal’s triangle. In general, the nth row consists of the binomial coefficients $\left(\begin{array}{c}n\\ r\end{array}\right)$, or ⁿC_r, with 0≤r≤n. With the numbers set out in this fashion, it can be seen how the number $\left(\begin{array}{c}n+1\\ r\end{array}\right)$ is equal to the sum of the two numbers $\left(\begin{array}{c}n\\ r-1\end{array}\right)$ and $\left(\begin{array}{c}n\\ r\end{array}\right)$, which are situated above it to the left and right. For example, $\left(\begin{array}{c}7\\ 3\end{array}\right)$ equals 35, and is the sum of $\left(\begin{array}{c}6\\ 2\end{array}\right)$, which equals 15, and $\left(\begin{array}{c}6\\ 3\end{array}\right)$, which equals 20.

See also Singmaster’s conjecture.