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Question Number 118937 by harckinwunmy last updated on 20/Oct/20

What is the general rule   for factorising (a+b+c)^n ?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{general}\:\mathrm{rule}\: \\ $$$$\mathrm{for}\:\mathrm{factorising}\:\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)^{\mathrm{n}} ? \\ $$

Answered by mathmax by abdo last updated on 20/Oct/20

i think the question is developping  (a+b+c)^n  =Σ_(k=0) ^n  C_n ^k (a+b)^k c^(n−k)   =Σ_(k=0) ^n  C_n ^k (Σ_(p=0) ^k  C_k ^p  a^p  b^(k−p) )c^(n+k)   =Σ_(k=0) ^n  C_n ^n  Σ_(p=0) ^k  C_k ^(p )  a^p  b^(k−p)  c^(n+k)

$$\mathrm{i}\:\mathrm{think}\:\mathrm{the}\:\mathrm{question}\:\mathrm{is}\:\mathrm{developping} \\ $$$$\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)^{\mathrm{n}} \:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \left(\mathrm{a}+\mathrm{b}\right)^{\mathrm{k}} \mathrm{c}^{\mathrm{n}−\mathrm{k}} \\ $$$$=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \left(\sum_{\mathrm{p}=\mathrm{0}} ^{\mathrm{k}} \:\mathrm{C}_{\mathrm{k}} ^{\mathrm{p}} \:\mathrm{a}^{\mathrm{p}} \:\mathrm{b}^{\mathrm{k}−\mathrm{p}} \right)\mathrm{c}^{\mathrm{n}+\mathrm{k}} \\ $$$$=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{n}} \:\sum_{\mathrm{p}=\mathrm{0}} ^{\mathrm{k}} \:\mathrm{C}_{\mathrm{k}} ^{\mathrm{p}\:} \:\mathrm{a}^{\mathrm{p}} \:\mathrm{b}^{\mathrm{k}−\mathrm{p}} \:\mathrm{c}^{\mathrm{n}+\mathrm{k}} \\ $$

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