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Question-220878

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Question Number 220878 by Spillover last updated on 20/May/25
Answered by Rasheed.Sindhi last updated on 22/May/25
((n!)/((n−r)!r!))+((2×n!)/((n−r+1)!(r−1)!))+((n!)/((n−r+2)!(r−2)!)) ((n!)/((n−r)!r(r−1)(r−2)!))+((2×n!)/((n−r+1)(n−r)!(r−1)(r−2)!))+((n!)/((n−r+2)(n−r+1)(n−r)!(r−2)!)) ((n!)/((n−r)!(r−2)!))((1/(r(r−1)))+(2/((n−r+1)(r−1)))+(1/((n−r+2)(n−r+1)))) ((n!)/((n−r)!(r−2)!))((((n−r+2)(n−r+1)+2r(n−r+2)+r(r−1))/(r(r−1)(n−r+2)(n−r+1)))) ((n!)/((n−r)!(r−2)!))((((n−r)^2 +3(n−r)+2+2r(n−r)+4r+r^2 −r)/(r(r−1)(n−r+2)(n−r+1)))) ((n!)/((n−r)!(r−2)!))(((n^2 −2nr+r^2 +3n−3r+2+2nr−2r^2 +3r+r^2 )/(r(r−1)(n−r+2)(n−r+1)))) ((n!)/((n−r)!(r−2)!))(((n^2 +3n+2)/(r(r−1)(n−r+2)(n−r+1)))) ((n!)/((n−r)!(r−2)!))((((n+2)(n+1))/(r(r−1)(n−r+2)(n−r+1)))) (((n+2)!)/(r!(n−r+2)!)) (((n+2)!)/(r!(n+2−r)!))= (((n+2)),(( r)) )
$$\frac{{n}!}{\left({n}−{r}\right)!{r}!}+\frac{\mathrm{2}×{n}!}{\left({n}−{r}+\mathrm{1}\right)!\left({r}−\mathrm{1}\right)!}+\frac{{n}!}{\left({n}−{r}+\mathrm{2}\right)!\left({r}−\mathrm{2}\right)!} \\ $$$$\frac{{n}!}{\left({n}−{r}\right)!{r}\left({r}−\mathrm{1}\right)\left({r}−\mathrm{2}\right)!}+\frac{\mathrm{2}×{n}!}{\left({n}−{r}+\mathrm{1}\right)\left({n}−{r}\right)!\left({r}−\mathrm{1}\right)\left({r}−\mathrm{2}\right)!}+\frac{{n}!}{\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)\left({n}−{r}\right)!\left({r}−\mathrm{2}\right)!} \\ $$$$\frac{{n}!}{\left({n}−{r}\right)!\left({r}−\mathrm{2}\right)!}\left(\frac{\mathrm{1}}{{r}\left({r}−\mathrm{1}\right)}+\frac{\mathrm{2}}{\left({n}−{r}+\mathrm{1}\right)\left({r}−\mathrm{1}\right)}+\frac{\mathrm{1}}{\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)}\right) \\ $$$$\frac{{n}!}{\left({n}−{r}\right)!\left({r}−\mathrm{2}\right)!}\left(\frac{\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)+\mathrm{2}{r}\left({n}−{r}+\mathrm{2}\right)+{r}\left({r}−\mathrm{1}\right)}{{r}\left({r}−\mathrm{1}\right)\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)}\right) \\ $$$$\frac{{n}!}{\left({n}−{r}\right)!\left({r}−\mathrm{2}\right)!}\left(\frac{\left({n}−{r}\right)^{\mathrm{2}} +\mathrm{3}\left({n}−{r}\right)+\mathrm{2}+\mathrm{2}{r}\left({n}−{r}\right)+\mathrm{4}{r}+{r}^{\mathrm{2}} −{r}}{{r}\left({r}−\mathrm{1}\right)\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)}\right) \\ $$$$\frac{{n}!}{\left({n}−{r}\right)!\left({r}−\mathrm{2}\right)!}\left(\frac{{n}^{\mathrm{2}} −\mathrm{2}{nr}+{r}^{\mathrm{2}} +\mathrm{3}{n}−\mathrm{3}{r}+\mathrm{2}+\mathrm{2}{nr}−\mathrm{2}{r}^{\mathrm{2}} +\mathrm{3}{r}+{r}^{\mathrm{2}} }{{r}\left({r}−\mathrm{1}\right)\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)}\right) \\ $$$$\frac{{n}!}{\left({n}−{r}\right)!\left({r}−\mathrm{2}\right)!}\left(\frac{{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{2}}{{r}\left({r}−\mathrm{1}\right)\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)}\right) \\ $$$$\frac{{n}!}{\left({n}−{r}\right)!\left({r}−\mathrm{2}\right)!}\left(\frac{\left({n}+\mathrm{2}\right)\left({n}+\mathrm{1}\right)}{{r}\left({r}−\mathrm{1}\right)\left({n}−{r}+\mathrm{2}\right)\left({n}−{r}+\mathrm{1}\right)}\right) \\ $$$$\frac{\left({n}+\mathrm{2}\right)!}{{r}!\left({n}−{r}+\mathrm{2}\right)!}\:\:\: \\ $$$$\frac{\left({n}+\mathrm{2}\right)!}{{r}!\left({n}+\mathrm{2}−{r}\right)!}=\begin{pmatrix}{{n}+\mathrm{2}}\\{\:\:\:\:{r}}\end{pmatrix} \\ $$

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