Question

if nC12=nC6 then nC2 is
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Answer 1

Question :

$\tt If \: {}^{n}{C}_{12} = {}^{n}{C}_{6} \: then \: {}^{n}{C}_{2} = ?$

Formula :

$\tt \large{{}^{n}{C}_{r} = \frac{n!}{(n-r)!r!}}$

Solution :

$\tt {}^{n}{C}_{12} = {}^{n}{C}_{6}$

$\tt \implies \frac{n!}{(n-12)!12!} = \frac{n!}{(n-6)!6!}$

$\tt \implies \frac{\cancel{n!}}{(n-12)!12!} = \frac{\cancel{n!}}{(n-6)!6!}$

$\tt \implies \frac{1}{(n-12)!12!} = \frac{1}{(n-6)!6!}$

$\tt \implies$ (n-6)!6! = (n-12)!12!

$\tt \implies \tiny{ (n-6)(n-7)(n-8)(n-9)(n-10)(n-11)(n-12)!6! = (n-12)!12×11×10×9×8×7×6! }$

$\tt \implies \tiny{(n-6)(n-7)(n-8)(n-9)(n-10)(n-11) = 12 ×11×10×9×8×7}$

$\tt \implies$ n-6 = 12

$\tt \implies$ n = 12 + 6

$\tt \implies$ n = 18

n = 18

$\tt {}^{n}{C}_{2}$

$\tt \implies {}^{18}{C}_{2}$

$\tt \implies \frac{18!}{(18-2)!2!}$

$\tt \implies \frac{18!}{16!2!}$

$\tt \implies \frac{18×17×16!}{16!2!}$

$\tt \implies \frac{18×17}{2}$

$\tt \implies 9 × 17$

$\tt \implies 153$

$\tt {}^{n}{C}_{2} = 153$