The number of permutations of n disimilar things taken not more than 'r' at a time, when each thing may occur any number of times is

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

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

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^{n} P_{1} +^{n} P_{2} + . . . . . . +^{n} P_{r}

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

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Solution

The correct option is A $\frac{n (n^{r} - 1)}{n - 1}$
Permutation of size 1
$\to n$
Permutation of size 2
$\to n^{2}$
....
....
Permutation of size r
$\to n^{r}$
S $=$ $n + n^{2} + n^{3} + . . . . . . . . . . . . . . . . . . . . . . + n^{r}$
S $=$ $\frac{n (n^{r} - 1)}{n - 1}$

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Similar questions

n

r

List I	List II
A. The number of ways of answering one or more of $n$ questions	i. $\frac{^{n} P_{r}}{2 r}$
B. The number of ways of answering one or more of $n$ questions is when each question has an alternative is	ii. $2^{n} - 1$
C. The number of circular permutations of $n$ different things taken $r$ at a time is	iii. $\frac{^{n} P_{r}}{r}$
D. The number of circular permutations of $n$ things taken $r$ at a time none direction is	iv. $3^{n} - 1$
	v. $2^{n}$

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