Question

Determine whether the statement $C\left(n,r\right)=P\left(n,r\right)$ is sometimes, always, or never true. Explain your reasoning.

Short answer

The statement istrue when r = 0 or 1.

Step-by-step solution

Step 1. Given Information.

Given to check whether the statement $C\left(n,r\right)=P\left(n,r\right)$ is sometimes, always, or never true.

Step 2. Calculation.

When n objects are available and r are to be picked then the number of combinations is given by $C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}$

When n objects are available and r are to be picked then the number of permutations is given by $P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

When $C\left(n,r\right)=P\left(n,r\right)$,

$\begin{array}{l}C\left(n,r\right)=P\left(n,r\right)\\ \frac{n!}{\left(n-r\right)!r!}=\frac{n!}{\left(n-r\right)!}\\ \frac{1}{1r!}=\frac{1}{1}\\ r!=1\end{array}$

For the statement to be true, $r!=1$i.e., $r=0\text{or}1$.

Step 3. Conclusion.

Therefore, the statement is true when r = 0 or 1.