Question

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

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

Hint:

Use geometric progression sum formula when repetition is allowed.

Answer 1

The Correct Option is A

With repetition allowed, number of $k$-length permutations: $n^k$
So total permutations for $k = 1$ to $r$ is: \[ \sum_{k=1}^{r} n^k = n + n^2 + \cdots + n^r = \frac{n(n^r - 1)}{n - 1} \] But options suggest indexing from $k = 0$ to $r$: i.e., \[ n^1 + n^2 + \cdots + n^r = \frac{n(n^{r} - 1)}{n - 1} \quad \text{(Option 3)} \] However, question says “not more than $r$” and starts from 1 ⇒ correct is: \[ \boxed{\frac{n(n^{r} - 1)}{n - 1}} \quad \text{[Typo in option marked, intended Option (3)]} \]