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Q. If $^nC_{12}= ^nC_{6}$ then $^nC_{2}$=

KCETKCET 2005Permutations and Combinations

Solution:

Given that,
${ }^{n} C_{12}={ }^{n} C_{6}$
Or ${ }^{n} C_{n-12}={ }^{n} C_{6} $
$\Rightarrow n-12=6$
$\Rightarrow \, n=18 $
$\therefore \, { }^{n} C_{2} ={ }^{18} C_{2}=\frac{18 \times 17}{2 \times 1}$
$=153$