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Question Number 104134 by bemath last updated on 19/Jul/20
what is the coefficient x^(15) in the expansion of x^6 (1−x)^(11)
$${what}\:{is}\:{the}\:{coefficient}\:{x}^{\mathrm{15}} \\ $$$${in}\:{the}\:{expansion}\:{of}\:{x}^{\mathrm{6}} \left(\mathrm{1}−{x}\right)^{\mathrm{11}} \\ $$
Answered by bobhans last updated on 19/Jul/20
”x^6 (1−x)^(11) = x^6 Σ_(n = 0) ^(11) C _n^(11) (1)^n (−x)^(11−n) = x^6 {...+ C _2^(11) (−x)^(11−2) + ... } = x^6 {...+ ((11.10)/(2.1)) (−x^9 )+...} so coefficient x^(15) is −55 . ■
$$''\boldsymbol{{x}}^{\mathrm{6}} \left(\mathrm{1}−\boldsymbol{{x}}\right)^{\mathrm{11}} \:=\:\boldsymbol{{x}}^{\mathrm{6}} \underset{{n}\:=\:\mathrm{0}} {\overset{\mathrm{11}} {\sum}}{C}\:_{{n}} ^{\mathrm{11}} \:\left(\mathrm{1}\right)^{{n}} \:\left(−{x}\right)^{\mathrm{11}−{n}} \\ $$$$=\:{x}^{\mathrm{6}} \:\left\{…+\:{C}\:_{\mathrm{2}} ^{\mathrm{11}} \left(−{x}\right)^{\mathrm{11}−\mathrm{2}} \:+\:…\:\right\} \\ $$$$=\:{x}^{\mathrm{6}} \:\left\{…+\:\frac{\mathrm{11}.\mathrm{10}}{\mathrm{2}.\mathrm{1}}\:\left(−{x}^{\mathrm{9}} \right)+…\right\} \\ $$$${so}\:{coefficient}\:{x}^{\mathrm{15}} \:{is}\:−\mathrm{55}\:.\:\blacksquare\: \\ $$
Answered by mathmax by abdo last updated on 19/Jul/20
x^6 (1−x)^(11) =−x^6 (x−1)^(11) =−x^6 Σ_(k=0) ^(11 ) C_(11) ^k x^k (−1)^(11−k) =x^6 Σ_(k=0) ^(11) C_(11) ^k (−1)^k x^k =Σ_(k=0) ^(11) (−1)^(k ) C_(11) ^k x^(k+6) we get the coefficient of x^(15) ifk+6 =15 ⇒k =9 so tbe coefficent is λ =(−1)^9 C_(11) ^9 =−((11!)/(9!2!)) =−((11×10)/2) =−55.
$$\mathrm{x}^{\mathrm{6}} \left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{11}} \:=−\mathrm{x}^{\mathrm{6}} \left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{11}} \:=−\mathrm{x}^{\mathrm{6}} \sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{11}\:} \:\mathrm{C}_{\mathrm{11}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}} \left(−\mathrm{1}\right)^{\mathrm{11}−\mathrm{k}} \\ $$$$=\mathrm{x}^{\mathrm{6}} \:\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{11}} \:\mathrm{C}_{\mathrm{11}} ^{\mathrm{k}} \left(−\mathrm{1}\right)^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}} \:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{11}} \:\left(−\mathrm{1}\right)^{\mathrm{k}\:} \mathrm{C}_{\mathrm{11}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}+\mathrm{6}} \:\:\mathrm{we}\:\mathrm{get}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{15}} \\ $$$$\mathrm{ifk}+\mathrm{6}\:=\mathrm{15}\:\Rightarrow\mathrm{k}\:=\mathrm{9}\:\mathrm{so}\:\mathrm{tbe}\:\mathrm{coefficent}\:\mathrm{is}\:\lambda\:=\left(−\mathrm{1}\right)^{\mathrm{9}} \:\mathrm{C}_{\mathrm{11}} ^{\mathrm{9}} \:=−\frac{\mathrm{11}!}{\mathrm{9}!\mathrm{2}!} \\ $$$$=−\frac{\mathrm{11}×\mathrm{10}}{\mathrm{2}}\:=−\mathrm{55}. \\ $$

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