If $^{49} C_{3 r - 2} =^{49} C_{2 r + 1},$ find $^{'} r^{'} .$

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Solution

$^{n} C_{r} =^{n} C_{s}$ if either $r = s$ or $r + s = n .$
Thus $3 r - 2 = 2 r + 1 \Rightarrow r = 3$
or $3 r - 2 + 2 r + 1 = 49 \Rightarrow 5 r - 1 = 49 \Rightarrow r = 10$
$∴$ $r = 3, 10$

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