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Question Number 139332 by bemath last updated on 26/Apr/21
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{50}} \:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expression}\:\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{1000}} \:+\mathrm{2x}\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{999}} + \\ $$$$\mathrm{3x}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{998}} +...+\mathrm{1001x}^{\mathrm{1000}} \\ $$
Answered by mr W last updated on 26/Apr/21
$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{1001}} {\sum}}{nx}^{{n}−\mathrm{1}} \left(\mathrm{1}+{x}\right)^{\mathrm{1001}−{n}} \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{1001}} {\sum}}\underset{{k}=\mathrm{0}} {\overset{\mathrm{1001}−{n}} {\sum}}{n}\begin{pmatrix}{\mathrm{1001}−{n}}\\{{k}}\end{pmatrix}{x}^{{k}+{n}−\mathrm{1}} \\ $$$${x}^{{k}+{n}−\mathrm{1}} ={x}^{\mathrm{50}} \:\Rightarrow{k}+{n}−\mathrm{1}=\mathrm{50}\:\Rightarrow{k}=\mathrm{51}−{n}\geqslant\mathrm{0}\:\Rightarrow{n}\leqslant\mathrm{51} \\ $$$${coef}.\:{of}\:{x}^{\mathrm{50}} : \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{51}} {\sum}}{n}\begin{pmatrix}{\mathrm{1001}−{n}}\\{\mathrm{51}−{n}}\end{pmatrix} \\ $$
Commented by mr W last updated on 26/Apr/21
Commented by bemath last updated on 28/Apr/21
$$\mathrm{i}\:\mathrm{got}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{50}} \:\mathrm{in}\:\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{1002}} \\ $$$$=\:\mathrm{C}\:_{\mathrm{50}} ^{\mathrm{1002}} \:=\:\frac{\mathrm{1002}!}{\mathrm{50}!\:\mathrm{952}!} \\ $$
Commented by mr W last updated on 28/Apr/21
$${how}\:{did}\:{you}\:{get}? \\ $$