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Question Number 138317 by abenarhodym last updated on 12/Apr/21

Commented by abenarhodym last updated on 12/Apr/21

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Answered by mr W last updated on 12/Apr/21

 (((n+1)),(3) )− (((n−1)),(3) )  =(((n+1)n(n−1))/(3!))−(((n−1)(n−2)(n−3))/(3!))  =(((n−1)(n^2 +n−n^2 +5n−6))/6)  =(n−1)^2

$$\begin{pmatrix}{{n}+\mathrm{1}}\\{\mathrm{3}}\end{pmatrix}−\begin{pmatrix}{{n}−\mathrm{1}}\\{\mathrm{3}}\end{pmatrix} \\ $$$$=\frac{\left({n}+\mathrm{1}\right){n}\left({n}−\mathrm{1}\right)}{\mathrm{3}!}−\frac{\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)\left({n}−\mathrm{3}\right)}{\mathrm{3}!} \\ $$$$=\frac{\left({n}−\mathrm{1}\right)\left({n}^{\mathrm{2}} +{n}−{n}^{\mathrm{2}} +\mathrm{5}{n}−\mathrm{6}\right)}{\mathrm{6}} \\ $$$$=\left({n}−\mathrm{1}\right)^{\mathrm{2}} \\ $$

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