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Question-168493




Question Number 168493 by mokys last updated on 11/Apr/22
Answered by FelipeLz last updated on 12/Apr/22
 ((n),(r) )− (((n−1)),((r−1)) ) = ((n!)/(r!(n−r)!))−(((n−1)!)/((r−1)!(n−r)!)) = ((n!−r(n−1)!)/(r!(n−r)!)) = (((n−r)(n−1)!)/(r!(n−r)!)) = (((n−1)!)/(r!(n−r−1)!)) = (((n−1)!)/(r!((n−1)−r)!)) =  (((n−1)),((     r)) )
$$\begin{pmatrix}{{n}}\\{{r}}\end{pmatrix}−\begin{pmatrix}{{n}−\mathrm{1}}\\{{r}−\mathrm{1}}\end{pmatrix}\:=\:\frac{{n}!}{{r}!\left({n}−{r}\right)!}−\frac{\left({n}−\mathrm{1}\right)!}{\left({r}−\mathrm{1}\right)!\left({n}−{r}\right)!}\:=\:\frac{{n}!−{r}\left({n}−\mathrm{1}\right)!}{{r}!\left({n}−{r}\right)!}\:=\:\frac{\left({n}−{r}\right)\left({n}−\mathrm{1}\right)!}{{r}!\left({n}−{r}\right)!}\:=\:\frac{\left({n}−\mathrm{1}\right)!}{{r}!\left({n}−{r}−\mathrm{1}\right)!}\:=\:\frac{\left({n}−\mathrm{1}\right)!}{{r}!\left(\left({n}−\mathrm{1}\right)−{r}\right)!}\:=\:\begin{pmatrix}{{n}−\mathrm{1}}\\{\:\:\:\:\:{r}}\end{pmatrix} \\ $$

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