If $^{2 n} C_{r} :^{n} C_{2} = 44 : 3$ , then for which of the following values of $r$ the value of $^{n} C_{r}$ will be $15$

r = 3

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r = 4

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r = 6

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r = 5

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Solution

The correct option is D $r = 4$
$\frac{^{2 n} C_{3}}{^{n} C_{2}} = \frac{44}{3}$

$\Rightarrow$ $\frac{\frac{2 n (2 n - 1) (2 n - 2)}{3 \times 2}}{\frac{n (n - 1)}{2}} = \frac{44}{3}$

$\Rightarrow$ $\frac{\frac{2 n (n - 1) \times 2 (n - 1)}{3 \times 2}}{\frac{n (n - 1)}{2}} = \frac{44}{3}$

$\Rightarrow$ $\frac{2 (2 n - 1) \times 2}{3} = \frac{44}{3}$

$\Rightarrow$ $4 (2 n - 1) = 44$
$\Rightarrow$ $2 n - 1 = 11$
$\Rightarrow$ $2 n = 12$
$∴$ $n = 6$

Now we need to find value of $r$ where, $^{6} C_{r} = 15$

Since, it is small value, we can easily see that,
$^{6} C_{2} = 15$
Since, $^{n} C_{r} =^{n} C_{n - r}$
So, $^{6} C_{2} =^{6} C_{4} = 15$

$∴$ $r = 2$ or $r = 4$

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