Chapter 1.2, Problem 10E

Pascal’s triangle gives a method for calculating the binomial coefficients: it begins as follows:

The nth row of this triangle gives the coefficients for ${\left(a+b\right)}^{n-1}$.

To find an entry in the table other than a 1 on the boundary, add the two nearest numbers in the row directly above. The equation $\left(\begin{array}{c}n\\ r\end{array}\right)=\left(\begin{array}{c}n-1\\ r\end{array}\right)+\left(\begin{array}{c}n-1\\ r-1\end{array}\right)$ called Pascal’s equation, explains why Pascal’s triangle works. Prove that this equation is correct.