How can Pascal’s triangle be used to find binomial coefficients?

Pascal’s triangle provides a visual method for determining binomial coefficients, which are the numerical coefficients found in the expansion of the binomial theorem.

To find a specific coefficient, you locate the $r$th entry of the $n$th row, where counting for both $n$ and $r$ begins at zero. The construction of the triangle follows a simple rule: each interior entry is the sum of the two entries directly above it.

For example, the rows represent the coefficients for $(a+b{)}^n$:
* $n=0$: 1
* $n=1$: 1, 1
* $n=2$: 1, 2, 1
* $n=3$: 1, 3, 3, 1
* $n=4$: 1, 4, 6, 4, 1

In combinatorics, these values correspond to the number of ways to choose $k$ objects from a set of $n$ objects, denoted as ${}_n{C}_k$. While named after Blaise Pascal, this array was known to mathematicians such as Al-Karaji and Jia Xian centuries earlier.12