If ^8 C_ r - ^7 C_ 3 = ^7 C_ 2 , Find r. - Vidyadip

Question

If ${^8}\text{C}_{\text{r}}-{^7}\text{C}_{\text{3}}={^7}\text{C}_{\text{2}},$ Find r.

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Answer

Applying formula ${^\text{n}\text{C}_{\text{r}}}=\frac{\text{n}!}{\text{r!}(\text{n}-\text{r})!}$
$\frac{8!}{\text{r!}(8-\text{r})!}=\frac{7!}{2!5!}+\frac{7!}{3!4!}$
$\frac{8\times7!}{\text{r}!(8-\text{r})!}=\frac{7!}{2\times5\times4!}+\frac{7!}{3\times2\times4!}$
$\frac{8\times7!}{\text{r}!(8-\text{r})!}=\frac{7!}{2\times5\times4!}\Big(\frac{1}{5}+\frac{1}{5}\Big)$
Cancelling from both
$\frac{8\times7!}{\text{r}!(8-\text{r})!}=\frac{8}{2\times15\times4!}$
Cancelling 8 from both sides
2 × 5 × 3 × 4 × 3 × 2 × 1 = r! (8 - r)!
⇒ r = 3

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