Question:
If 8Cr − 7C3 = 7C2, find r.
Solution:
Given:
8Cr − 7C3 = 7C2
We have,
${ }^{8} C_{r}={ }^{7} C_{2}+{ }^{7} C_{3}$
$\Rightarrow{ }^{8} C_{r}={ }^{8} C_{3} \quad\left[\because{ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r} ; r \leq n\right]$
$\Rightarrow r=3 \quad\left[\because{ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y\right.$ or, $\left.n=x+y\right]$
And $r+3=8$
$\Rightarrow r=5$
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