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If $\mathrm{C}(20, \mathrm{n}+2)=\mathrm{C}(20, \mathrm{n}-2)$, then what is $\mathrm{n}$ equal to ?
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The correct answer is:
10
Given, $\mathrm{C}(20, \mathrm{n}+2)=\mathrm{C}(20, \mathrm{n}-2)$
$\Rightarrow{ }^{20} \mathrm{C}_{\mathrm{n}+2}={ }^{20} \mathrm{C}_{\mathrm{n}-2}$
$\Rightarrow 20=\mathrm{n}+2+\mathrm{n}-2 \quad\left(\because{ }^{n} c_{r}={ }^{n} c_{s} \Rightarrow n=r+s\right)$
$\Rightarrow 20=2 \mathrm{n}$
$\Rightarrow \mathrm{n}=10$
$\Rightarrow{ }^{20} \mathrm{C}_{\mathrm{n}+2}={ }^{20} \mathrm{C}_{\mathrm{n}-2}$
$\Rightarrow 20=\mathrm{n}+2+\mathrm{n}-2 \quad\left(\because{ }^{n} c_{r}={ }^{n} c_{s} \Rightarrow n=r+s\right)$
$\Rightarrow 20=2 \mathrm{n}$
$\Rightarrow \mathrm{n}=10$
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