Which among the following is equal to $^{n} C_{r}$ ?

^{n} C_{n - r}

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^{n} C_{n}

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

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^{r} C_{n}

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Solution

The correct option is C $\frac{n!}{r! (n - r)!}$

$^{n} C_{r}$ can be expressed as:
$^{n} C_{r} = \frac{n!}{r! (n - r)!}$

Replacing $r$ by $n - r$ in the above formula:
$^{n} C_{n - r} = \frac{n!}{(n - r)! (n - (n - r))!}$

$^{n} C_{n - r} = \frac{n!}{(n - r)! (n - n + r)!}$

$^{n} C_{r} = \frac{n!}{(n - r)! r!}$

$^{n} C_{r} = \frac{n!}{(n - r)! r!} =^{n} C_{n - r}$

Therefore, option (a.) and (c.) are the correct answers.

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Similar questions

^{n} C_{r + 1} +^{n} C_{r - 1} + 2 \times^{n} C_{r}

^{n} C_{r} + 2^{n} C_{r - 1} +^{n} C_{r - 2}

^{n} C_{r - 1} = 56

^{n} C_{r} = 28

^{n} C_{r + 1} = 8

^{n} C_{r - 1} = 10,^{n} C_{r} = 45

^{n} C_{r + 1} = 120

r

^{n} C_{r} +^{n} C_{r + 1} =^{n + 1} C_{x}

x

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