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Question Number 10853 by geovane10math last updated on 27/Feb/17

(x + y)^n  = Σ_(k=0) ^n  ((n),(k) )x^k y^(n−k)   (x − y)^n  = ???????

$$\left({x}\:+\:{y}\right)^{{n}} \:=\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}{x}^{{k}} {y}^{{n}−{k}} \\ $$$$\left({x}\:−\:{y}\right)^{{n}} \:=\:??????? \\ $$

Answered by mrW1 last updated on 27/Feb/17

(x − y)^n  = Σ_(k=0) ^n  ((n),(k) )(−1)^(n−k) x^k y^(n−k)

$$\left({x}\:−\:{y}\right)^{{n}} \:=\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}\left(−\mathrm{1}\right)^{{n}−{k}} {x}^{{k}} {y}^{{n}−{k}} \\ $$

Commented by geovane10math last updated on 27/Feb/17

Thanks!

$$\mathrm{Thanks}! \\ $$

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