The number of permutations of n different things taking r at a time when 3 particluar things are to be included is

$^{n - 3} P_{r - 3}$

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$^{n - 3} P_{r}$

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$^{n} P_{r - 3}$

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$r!^{n - 3} C_{r - 3}$

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Solution

The correct option is D
$r!^{n - 3} C_{r - 3}$

Here, we have to permute n things of which 3 things are to be included.

So, only the remaining (n-3) things are left for permutation, taking (r-3) things at a time. This is because 3 things have already been included.

But, these r things can be arranged in r! ways

$∴$ Total number of permutations

$= r!^{n - 3} C_{r - 3}$

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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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