First slide

Permutations

Question

The total number of permutations of $n (> 1)$ different things taken not more that $r$ at a time, when
a thing may be repeated any number of times, is

Moderate

A

$\frac{n}{n - 1} (n^{r} - 1)$
B

$\frac{n^{r - 1}}{n - 1}$
C

$\frac{n^{r} + 1}{n + 1}$
D

$\frac{n^{r} + n}{n - 1}$

Solution

When $k (1 \leq k \leq r)$ things are arranged, the number of possible arrangements, when repetition is allowed is
$\underset{k times}{\underset{⏟}{n \times n \times \dots \dots \times n}} = n^{k}$
Thus, the total possible arrangements is
$n + n^{2} + n^{3} + \dots + n^{r} = \frac{n (n^{r} - 1)}{n - 1}$

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