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Question Number 34878 by NECx last updated on 12/May/18
show that C_(n−1) ^(n+r−1) =C_r ^(n+r−1)
$${show}\:{that} \\ $$$$\overset{{n}+{r}−\mathrm{1}} {{C}}_{{n}−\mathrm{1}} =\overset{{n}+{r}−\mathrm{1}} {{C}}_{{r}} \\ $$
Answered by Rasheed.Sindhi last updated on 12/May/18
^n C_r =((n!)/(r!(n−r)!)) C_(n−1) ^(n+r−1) =C_r ^(n+r−1) LHS:(((n+r−1)!)/((n−1)!(n+r−1− n−1 ^(−) )!)) (((n+r−1)!)/((n−1)!(n+r−1− n+1 )!)) (((n+r−1)!)/(r!(n−1)! )) RHS:(((n+r−1)!)/(r!(n+r−1−r)!)) (((n+r−1)!)/(r!(n−1)!)) LHS=RHS Proved
$$\:^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} =\frac{\mathrm{n}!}{\mathrm{r}!\left(\mathrm{n}−\mathrm{r}\right)!} \\ $$$$\overset{{n}+{r}−\mathrm{1}} {{C}}_{{n}−\mathrm{1}} =\overset{{n}+{r}−\mathrm{1}} {{C}}_{{r}} \\ $$$$\mathrm{LHS}:\frac{\left(\mathrm{n}+\mathrm{r}−\mathrm{1}\right)!}{\left(\mathrm{n}−\mathrm{1}\right)!\left(\mathrm{n}+\mathrm{r}−\mathrm{1}−\overline {\:\mathrm{n}−\mathrm{1}\:}\right)!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\frac{\left(\mathrm{n}+\mathrm{r}−\mathrm{1}\right)!}{\left(\mathrm{n}−\mathrm{1}\right)!\left(\mathrm{n}+\mathrm{r}−\mathrm{1}−\:\mathrm{n}+\mathrm{1}\:\right)!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\frac{\left(\mathrm{n}+\mathrm{r}−\mathrm{1}\right)!}{\mathrm{r}!\left(\mathrm{n}−\mathrm{1}\right)!\:} \\ $$$$\mathrm{RHS}:\frac{\left(\mathrm{n}+\mathrm{r}−\mathrm{1}\right)!}{\mathrm{r}!\left(\mathrm{n}+\mathrm{r}−\mathrm{1}−\mathrm{r}\right)!}\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\frac{\left(\mathrm{n}+\mathrm{r}−\mathrm{1}\right)!}{\mathrm{r}!\left(\mathrm{n}−\mathrm{1}\right)!}\:\:\:\:\:\:\: \\ $$$$\mathrm{LHS}=\mathrm{RHS} \\ $$$$\:\:\mathrm{Proved} \\ $$

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