Chapter 13.2, Problem 27HP

To determine

Toprove:that the given equation $C=C{}_r$ is correct.

Answer to Problem 27HP

The equation is proved $C=C{}_r$ .

Explanation of Solution

Given:

  $C=C{}_r$ .

Concept used:

According to the combination formula:

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

According to the permutation formula:

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

Calculation:

Set the two sides equal to each other. Develop each side using the formulas in this lesson and simplify. Working with the left side will eventually reduce it to the right side.

  $\begin{array}{l}C=C{}_r.\\ \frac{n!}{\left(n-\left(n-r\right)!\right)\left(n-r\right)!}=\frac{n!}{\left(n-r\right)!r!}.\\ \frac{n!}{\left(n-r\right)!r!}=\frac{n!}{\left(n-r\right)!r!}.\end{array}$

Hence. The equation is proved $C=C{}_r$ .