Menu Close

Prove-that-the-greatest-coefficient-in-the-expansion-of-x-1-x-2-x-3-x-k-n-n-q-k-r-q-1-r-where-n-qk-r-0-r-k-1-




Question Number 22315 by Tinkutara last updated on 15/Oct/17
Prove that the greatest coefficient in  the expansion of (x_1 +x_2 +x_3 +...+x_k )^n   = ((n!)/((q!)^(k−r) [(q+1)!]^r )) , where n = qk + r,  0 ≤ r ≤ k − 1
$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{coefficient}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left({x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +…+{x}_{{k}} \right)^{{n}} \\ $$$$=\:\frac{{n}!}{\left({q}!\right)^{{k}−{r}} \left[\left({q}+\mathrm{1}\right)!\right]^{{r}} }\:,\:\mathrm{where}\:{n}\:=\:{qk}\:+\:{r}, \\ $$$$\mathrm{0}\:\leqslant\:{r}\:\leqslant\:{k}\:−\:\mathrm{1} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *