Question

The total number of permutation of $n$ different things taken not more than $r$ at a time, where each thing may be repeated any number of times, is

• A. $\frac{n(n^n-1)}{n-1}$
• B. $\frac{(n^r-1)}{n-1}$
• C. $\frac{n(n^r-1)}{n-1}$
• D. $\frac{(n^n-1)}{n-1}$

Answer 1

The correct option is C $\frac{n(n^r-1)}{n-1}$

One place can be filled in $n$ ways by $n$ things, the first and second places can be filled in $=n\times n=n^2$ ways.

The first, second and third places can be filled in $=n\times n\times n=n^3$

Similarly, the first, second,....rth places can be filled in $=n\times n\times n\times ......\times n\left(rtimes\right)=n^r$

So required total number $=n+n^2+n^3+.....+n^r=\frac{n(n^r-1)}{n-1}$ (Sum of a GP)