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Permutations and Combinations

If r 2-13 r +...

Question

If $r^{2} - 13 r + 40 = 0$ , then the value of $^{7} C_{r}$ is

A

56

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B

12

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C

1

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D

21

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Solution

The correct option is D $21$
Given : $r^{2} - 13 r + 40 = 0$
$\Rightarrow (r - 8) (r - 5) = 0$
$\Rightarrow r = 8, 5$
We know, $n \geq r$ for $^{n} C_{r}$
So, only possible value of $r = 5$
$^{7} C_{r} =^{7} C_{5}$
$\Rightarrow^{7} C_{5} = \frac{7!}{2! \cdot 5!} = \frac{7 \cdot 6 \cdot 5!}{2 \cdot 1 \cdot 5!}$
$∴^{7} C_{5} = 21$

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