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  4. If nC2 + nC3 = 6C3 and nCx = nC3, x ≠ 3, then the value of x is equal to

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Q. If $^n$C$_2$ + $^n$C$_3$ = $^6$C$_3$ and $^n$C$_x$ = $^n$C$_3$, x ≠ 3, then the value of x is equal to

KEAMKEAM 2016

Solution:

Given, ${ }^{n} C_{2}+{ }^{n} C_{3}={ }^{6} C_{3}$
$\Rightarrow n=5 \left[\because{ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}\right]$
and ${ }^{n} C_{x}={ }^{n} C_{3} \Rightarrow { }^{5} C_{x}={ }^{5} C_{3}$
$\Rightarrow x =5-3=2$
$\left[\because\right.$ If ${ }^{n} C_{r 1}={ }^{n} C_{12} \Rightarrow $ either $r_{1}=r_{2}$ or $\left.n=r_{1}+r_{2}\right]$