Which of the following is correct number of ways for selecting $n$ different things taken $k$ at a time?

\frac{n!}{(n - k)!}

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\frac{n!}{(n - k)! \times k!}

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^{n} C_{k}

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\frac{^{n} P_{k}}{k!}

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Solution

The correct option is D $\frac{^{n} P_{k}}{k!}$
The number of ways of selectinh $n$ different things taken $k$ at a time is $^{n} C_{k} .$

We can also write $^{n} C_{k} = \frac{n!}{(n - k)! \times k!}$

We know that $^{n} P_{k} = \frac{n!}{(n - k)!}$

On dividing $^{n} P_{k}$ by $k!$ we get,
$\frac{^{n} P_{k}}{k!} = \frac{n!}{(n - k)!} \times \frac{1}{k!}$

$\Rightarrow \frac{^{n} P_{k}}{k!} = \frac{n!}{(n - k)! k!} =^{n} C_{k} .$

So, option $(b), (c), (d)$ are correct.

Suggest Corrections

Similar questions

^{'} n^{'}

^{'} r^{'}

= \frac{n!}{(n - r)!}

Statement 2: Number of ways if Order does not matter

= \frac{n!}{(n - r)! \times r!}

2 n

^{'} n^{'}

^{'} n^{'}

^{'} n^{'}

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. $2^{n}$

n

3

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