Question Number 2196 by Rasheed Soomro last updated on 08/Nov/15
$$\left({a}+{b}\right)^{\mathrm{2}} ={a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\mathrm{2}{ab} \\ $$$$\left({a}+{b}+{c}\right)^{\mathrm{2}} ={a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\mathrm{2}{ab}+\mathrm{2}{bc}+\mathrm{2}{ca} \\ $$$$\left({a}+{b}+{c}+{d}\right)^{\mathrm{2}} =? \\ $$$$\left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} +…+{a}_{{n}} \right)^{\mathrm{2}} =? \\ $$$${Derive}\:{a}\:{formula}\:{or}\:{give}\:{a}\:{technique}. \\ $$
Answered by 123456 last updated on 08/Nov/15
$$\left({a}_{\mathrm{1}} +\centerdot\centerdot\centerdot+{a}_{{n}} \right)^{{k}} =\underset{\alpha_{\mathrm{1}} \centerdot\centerdot\centerdot\alpha_{{n}} } {\sum}\begin{pmatrix}{{k}}\\{\alpha_{\mathrm{1}} ,\centerdot\centerdot\centerdot\alpha_{{n}} }\end{pmatrix}{a}_{\mathrm{1}} ^{\alpha_{\mathrm{1}} } \centerdot\centerdot\centerdot{a}_{{n}} ^{\alpha_{{n}} } \\ $$$$\Sigma\alpha_{{n}} ={k},\alpha_{{i}} \in\mathbb{N},{k}\in\mathbb{N} \\ $$$$\begin{pmatrix}{\alpha}\\{\alpha_{\mathrm{1}} ,\centerdot\centerdot\centerdot,\alpha_{{n}} }\end{pmatrix}=\frac{\alpha!}{\alpha_{\mathrm{1}} !\centerdot\centerdot\centerdot\alpha_{{n}} !} \\ $$
Commented by 123456 last updated on 08/Nov/15
$$\mathrm{multinomial}\:\mathrm{coeficients}\:\mathrm{and}\:\mathrm{multinomil} \\ $$$$\mathrm{expansion} \\ $$
Commented by Rasheed Soomro last updated on 08/Nov/15
$${What}\:{are}\:\alpha_{{i}} ?\:{For}\:{example}\:{for}\:\left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} +{a}_{\mathrm{3}} +{a}_{\mathrm{4}} \right)^{\mathrm{3}} \:{what} \\ $$$${are}\:\alpha_{{i}} \:\:? \\ $$
Commented by 123456 last updated on 08/Nov/15
$$\mathrm{the}\:\mathrm{possible}\:\mathrm{natural}\:\mathrm{pars}\:\mathrm{such}\:\mathrm{that} \\ $$$$\alpha_{\mathrm{1}} +\centerdot\centerdot\centerdot+\alpha_{{n}} ={k} \\ $$$$\alpha_{{i}} \in\mathbb{N} \\ $$