Question and Answers Forum

All Questions      Topic List

Permutation and Combination Questions

Previous in All Question      Next in All Question      

Previous in Permutation and Combination      Next in Permutation and Combination      

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

$$\:^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} =\frac{\mathrm{n}!}{\mathrm{r}!\left(\mathrm{n}−\mathrm{r}\right)!} \\ $$$$\overset{{n}+{r}−\mathrm{1}} {{C}}_{{n}−\mathrm{1}} =\overset{{n}+{r}−\mathrm{1}} {{C}}_{{r}} \\ $$$$\mathrm{LHS}:\frac{\left(\mathrm{n}+\mathrm{r}−\mathrm{1}\right)!}{\left(\mathrm{n}−\mathrm{1}\right)!\left(\mathrm{n}+\mathrm{r}−\mathrm{1}−\overline {\:\mathrm{n}−\mathrm{1}\:}\right)!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\frac{\left(\mathrm{n}+\mathrm{r}−\mathrm{1}\right)!}{\left(\mathrm{n}−\mathrm{1}\right)!\left(\mathrm{n}+\mathrm{r}−\mathrm{1}−\:\mathrm{n}+\mathrm{1}\:\right)!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\frac{\left(\mathrm{n}+\mathrm{r}−\mathrm{1}\right)!}{\mathrm{r}!\left(\mathrm{n}−\mathrm{1}\right)!\:} \\ $$$$\mathrm{RHS}:\frac{\left(\mathrm{n}+\mathrm{r}−\mathrm{1}\right)!}{\mathrm{r}!\left(\mathrm{n}+\mathrm{r}−\mathrm{1}−\mathrm{r}\right)!}\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\frac{\left(\mathrm{n}+\mathrm{r}−\mathrm{1}\right)!}{\mathrm{r}!\left(\mathrm{n}−\mathrm{1}\right)!}\:\:\:\:\:\:\: \\ $$$$\mathrm{LHS}=\mathrm{RHS} \\ $$$$\:\:\mathrm{Proved} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com