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Category: Permutation and Combination

If-n-is-positive-integer-prove-that-the-cofficient-of-x-2-and-x-3-in-the-expansion-of-x-2-2x-2-n-are-2-n-1-n-2-and-2-n-1-n-n-1-1-3-

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Question Number 9573 by lepan last updated on 17/Dec/16 $${If}\:{n}\:{is}\:{positive}\:{integer}\:{prove}\:{that}\: \\ $$$${the}\:{cofficient}\:{of}\:{x}^{\mathrm{2}\:} {and}\:{x}^{\mathrm{3}} \:{in}\:{the}\: \\ $$$${expansion}\:{of}\:\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}\right)^{{n}} \:{are}\:\mathrm{2}^{{n}−\mathrm{1}} .{n}^{\mathrm{2}} \\ $$$${and}\:\mathrm{2}^{{n}−\mathrm{1}} {n}\left({n}−\mathrm{1}\right)\frac{\mathrm{1}}{\mathrm{3}}. \\ $$ Commented…

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Find-the-coefficient-of-x-50-in-the-expression-1-x-1000-2x-1-x-999-3x-2-1-x-998-1001x-1000-

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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…

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1-1-x-n-n-0-n-1-x-n-2-x-2-n-3-x-3-

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Question Number 139286 by qaz last updated on 25/Apr/21 $$\frac{\mathrm{1}}{\left(\mathrm{1}−{x}\right)^{{n}} }=\left(\begin{pmatrix}{{n}}\\{\mathrm{0}}\end{pmatrix}\right)+\left(\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}\right){x}+\left(\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}\right){x}^{\mathrm{2}} +\left(\begin{pmatrix}{{n}}\\{\mathrm{3}}\end{pmatrix}\right){x}^{\mathrm{3}} +… \\ $$ Commented by mr W last updated on 25/Apr/21 $${wrong}! \\…

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