If $^{n} C_{r - 1} +^{n + 1} C_{r - 1} +^{n + 2} C_{r - 1} + . . . . . . . +^{2 n} C_{r - 1}, =^{2 n + 1} C_{r^{2} - 132} -^{n} C_{r}$ , then the value of $r$

10

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12

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Solution

The correct option is C $12$
$^{n} C_{r - 1} +^{n + 1} C_{r - 1} +^{n + 2} C_{r - 1} + . . . . . . . . . . . . . . . +^{2 n} C_{r - 1} =^{2 n + 1} C_{r^{2} - 132} -^{n} C_{r}$
$\Rightarrow^{n} C_{r} +^{n} C_{r - 1} +^{n + 1} C_{r - 1} +^{n + 2} C_{r - 1} + . . . . . . . . . . . . . . . +^{2 n} C_{r - 1} =^{2 n + 1} C_{r^{2} - 132}$
$\Rightarrow^{2 n + 1} C_{r} =^{2 n + 1} C_{r^{2} - 132}$
$\Rightarrow r = r^{2} - 132$
$\Rightarrow r (r - 1) = 132 = 12 \times 11$
$\Rightarrow r = 12$

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