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Permutations and Combinations

If 5 C r = r ...

Question

If $^{5} C_{r} = r \cdot^{5} C_{r - 1}$ , then the number of value(s) of $r$ is

A

2

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B

1

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C

0

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D

3

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Solution

The correct option is B $1$
Given : $^{5} C_{r} = r \cdot^{5} C_{r - 1}$
$\Rightarrow \frac{^{5} C_{r}}{^{5} C_{r - 1}} = r$
$\Rightarrow \frac{5 - r + 1}{r} = r$ $(since \frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r})$

$∴ r^{2} + r - 6 = 0$
$\Rightarrow (r + 3) (r - 2) = 0$
$r = - 3, 2$
rejecting $r = - 3$ as $r \in W$
$∴$ Number of solution $= 1$

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Permutations and Combinations

Standard XII Mathematics

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