Question Number 66262 by peter frank last updated on 11/Aug/19
![show that ^n c_(r+1) +^n c_(r ) =^(n+1) c_(r+1)](https://www.tinkutara.com/question/Q66262.png)
$$\boldsymbol{{show}}\:\boldsymbol{{that}}\: \\ $$$$\:^{\boldsymbol{{n}}} \boldsymbol{{c}}_{\boldsymbol{{r}}+\mathrm{1}} +^{\boldsymbol{{n}}} \boldsymbol{{c}}_{\boldsymbol{{r}}\:\:} =^{\boldsymbol{{n}}+\mathrm{1}} \boldsymbol{{c}}_{\boldsymbol{{r}}+\mathrm{1}} \\ $$
Commented by mathmax by abdo last updated on 12/Aug/19
![C_n ^r +C_n ^(r+1) =((n!)/(r!(n−r)!)) +((n!)/((r+1)!(n−r−1)!)) =((n!(r+1))/((r+1)!(n−r)!)) +((n!(n−r))/((r+1)!(n−r)!)) =((n!)/((r+1)!(n−r)!)){(r+1) +(n−r)} =(((n+1)!)/((r+1)!((n+1)−(r+1))!)) =C_(n+1) ^(r+1)](https://www.tinkutara.com/question/Q66265.png)
$${C}_{{n}} ^{{r}} \:+{C}_{{n}} ^{{r}+\mathrm{1}} \:=\frac{{n}!}{{r}!\left({n}−{r}\right)!}\:+\frac{{n}!}{\left({r}+\mathrm{1}\right)!\left({n}−{r}−\mathrm{1}\right)!} \\ $$$$=\frac{{n}!\left({r}+\mathrm{1}\right)}{\left({r}+\mathrm{1}\right)!\left({n}−{r}\right)!}\:+\frac{{n}!\left({n}−{r}\right)}{\left({r}+\mathrm{1}\right)!\left({n}−{r}\right)!} \\ $$$$=\frac{{n}!}{\left({r}+\mathrm{1}\right)!\left({n}−{r}\right)!}\left\{\left({r}+\mathrm{1}\right)\:+\left({n}−{r}\right)\right\}\:=\frac{\left({n}+\mathrm{1}\right)!}{\left({r}+\mathrm{1}\right)!\left(\left({n}+\mathrm{1}\right)−\left({r}+\mathrm{1}\right)\right)!} \\ $$$$={C}_{{n}+\mathrm{1}} ^{{r}+\mathrm{1}} \: \\ $$
Commented by peter frank last updated on 12/Aug/19
![thank you](https://www.tinkutara.com/question/Q66266.png)
$${thank}\:{you} \\ $$
Commented by mathmax by abdo last updated on 12/Aug/19
![you are welcome.](https://www.tinkutara.com/question/Q66268.png)
$${you}\:{are}\:{welcome}. \\ $$