Chapter 12, Problem 22CT

To determine

Verify that $r$ to be greater than $n$ .

Answer to Problem 22CT

It is not possible.

Explanation of Solution

Given information:

  ${}_n{C}_r$

Calculation:

Use the definition of combination.

Combination is defined as, the selection or combination of objects without any order. For any positive integer’s $n,and,r,$ the selection of $n$ objects taking $r$ at time, where $r\le n,$ is denoted by ${}_n{C}_r$ and defined by;

  ${}_n{C}_r=\frac{n!}{r!(n-r)!}\mathrm{......}\left(1\right)$

Consider $r>n$ , then the value of the $(n-r)!$ is negative. Which contradict the definition of factorial.

Factorial is only defined for positive integers. So, the value of $(n-r)!$ cannot find if $r>n$ .

Hence, it is not possible to define ${}_n{C}_r$ when $r$ to greater than $n$ .