Chapter 6.1, Problem 47E

Solution

To verify: the given identity.

  $C{}_r=C{}_r+C{}_{r-1}$.

Given:

  $C{}_r=C{}_r+C{}_{r-1}$.

Formula used:

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

Calculation:

Consider,

  $\begin{array}{l}C{}_r=\frac{\left(n+1\right)!}{r!\left[\left(n+1\right)-r\right]!}\\ C{}_r=\frac{\left(n+1\right)!}{r!\left(n-r+1\right)!}\to \left(1\right)\end{array}$

And,

From the equations (1) and (2), it follows that

  $C{}_r=C{}_r+C{}_{r-1}$