Let $P (m, n)$ be the foot of the perpendicular from the point $Q (2, 3)$ on the line $x + y - 11 = 0 .$ If $r$ and $s$ are non-negative integers, then the maximum value of $^{m} C_{r} +^{n} C_{s}$ is

20

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30

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40

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25

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Solution

The correct option is B $30$
Given, $P (m, n)$ is the foot of the perpendicular from $Q (2, 3)$ on the line $x + y - 11 = 0.$
$∴ \frac{m - 2}{1} = \frac{n - 3}{1} = \frac{- (2 + 3 - 11)}{1^{2} + 1^{2}} = 3$
$\Rightarrow m = 5, n = 6$

$^{m} C_{r} +^{n} C_{s}$
$=^{5} C_{r} +^{6} C_{s}$
$^{5} C_{r}$ is maximum when $r = \frac{5 + 1}{2} = 3$
and $^{6} C_{s}$ is maximum when $s = \frac{6}{2} = 3$
$∴$ Maximum value of $^{m} C_{r} +^{n} C_{s}$ is
$^{5} C_{3} +^{6} C_{3}$
$= 10 + 20 = 30$

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