Verify that $^{8} C_{4} +^{8} C_{5} =^{9} C_{4}$

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Solution

$^{8} C_{4} +^{8} C_{5} =^{9} C_{4} L H S =^{8} C_{4} +^{8} C_{5} \Rightarrow \frac{8!}{4! 4!} + \frac{8!}{5! 3!} \Rightarrow \frac{8!}{4! 3!} [\frac{1}{4} + \frac{1}{5}] \Rightarrow \frac{8!}{4! 3!} [\frac{5 + 4}{4! 5!}] \Rightarrow \frac{9!}{5! 4!} =^{9} C_{4} = R H S$

Hence verified as $L H S = R H S$

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