Chapter 5, Problem 4P

In the following triangular table, known as Pascal’s triangle, the entries in the nth row are the binomial coefficients $\left(\begin{array}{l}n\\ 0\end{array}\right),\left(\begin{array}{l}n\\ 1\end{array}\right),\left(\begin{array}{l}n\\ 2\end{array}\right),\dots ,\left(\begin{array}{l}n\\ n\end{array}\right)$.

Observe that each number (other than the ones) is the sum of the two numbers directly above it. For example, in the 5th row, the number 5 is the sum of the numbers 1 and 4 from the 4th row, and the number 10 is the sum of the numbers 4 and 6 from the 4th row. This fact is known as Pascal’s formula. Namely, the formula says that

$\left(\begin{array}{l}n\\ r\end{array}\right)=\left(\begin{array}{l}n-1\\ r-1\end{array}\right)+\left(\begin{array}{l}n-1\\ r\end{array}\right)$.

Use Pascal’s formula to extend Pascal’s triangle to the 12th row. Determine the values of $\left({}_5^{12}\right)$ and $\left({}_6^{12}\right)$ from the extended triangle.