Question

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

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

Answer 1

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

Here repetition of things is allowed and at most r can be taken at a time.
Now total number of things $=n$
The number of ways of takin 1 at a time $=n$
The number of ways of taking 2 at a time $=n^2$
The number of ways taking three at a time $=n^3$
Continuing the process to take r things at a time the number of ways are $=n^r$
$\therefore$ total no.of ways taking 1,2,...r at a time $=n+n^2+n^3+....+n^r$ $=n(1+n+n^2+....+n^{r-1})$
$=n(n^{r-1}+n^{r-2}+...+1)$
$=n\left[\frac{n^{r-1}+n^{r-2}+...+1}{n-1}\right]$$(n-1)$
multiplying and dividing by $n-1$
$=\frac{n(n^r-1)}{n-1}$