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Question Number 120016 by aurpeyz last updated on 28/Oct/20

Answered by mr W last updated on 28/Oct/20

S=(((1+x)^(16) −(1+x)^6 )/x) C_7 ^(16) =C_9 ^(16) ⇒answer (a)

$${S}=\frac{\left(\mathrm{1}+{x}\right)^{\mathrm{16}} −\left(\mathrm{1}+{x}\right)^{\mathrm{6}} }{{x}} \\ $$$${C}_{\mathrm{7}} ^{\mathrm{16}} ={C}_{\mathrm{9}} ^{\mathrm{16}} \\ $$$$\Rightarrow{answer}\:\left({a}\right) \\ $$

Answered by mathmax by abdo last updated on 29/Oct/20

(1+x)^6 +(1+x)^7 +....(1+x)^(15) =(1+x)^6 {1+(1+x)+(1+x)^2 +...+(1+x)^9 ) =(1+x)^6 ×((1−(1+x)^(10) )/(−x)) =(1+x)^6 ×(((1+x)^(10) −1)/x) =(((1+x)^(16) −(1+x)^6 )/x) =(1/x){ Σ_(k=0) ^(16) C_(16) ^k x^k −Σ_(k=0) ^6 C_6 ^k x^k } =(1/x)( Σ_(k=1) ^(16) C_(16) ^(k ) x^k −Σ_(k=1) ^6 C_6 ^k x^k )=Σ_(k=1) ^(16) C_(16) ^k x^(k−1) −Σ_(k=1) ^6 C_6 ^k x^(k−1) ⇒the coefficient is λ =C_(16) ^7 =((16!)/(7! 9!))

$$\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{6}} \:+\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{7}} +....\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{15}} \:=\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{6}} \left\{\mathrm{1}+\left(\mathrm{1}+\mathrm{x}\right)+\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} +...+\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{9}} \right) \\ $$$$=\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{6}} ×\frac{\mathrm{1}−\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{10}} }{−\mathrm{x}}\:=\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{6}} ×\frac{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{10}} −\mathrm{1}}{\mathrm{x}} \\ $$$$=\frac{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{16}} −\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{6}} }{\mathrm{x}}\:=\frac{\mathrm{1}}{\mathrm{x}}\left\{\:\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{16}} \:\mathrm{C}_{\mathrm{16}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}} −\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{6}} \:\mathrm{C}_{\mathrm{6}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}} \right\} \\ $$$$=\frac{\mathrm{1}}{\mathrm{x}}\left(\:\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{16}} \:\mathrm{C}_{\mathrm{16}} ^{\mathrm{k}\:} \:\mathrm{x}^{\mathrm{k}} −\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{6}} \:\mathrm{C}_{\mathrm{6}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}} \right)=\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{16}} \:\mathrm{C}_{\mathrm{16}} ^{\mathrm{k}} \mathrm{x}^{\mathrm{k}−\mathrm{1}} −\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{6}} \:\mathrm{C}_{\mathrm{6}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}−\mathrm{1}} \:\Rightarrow\mathrm{the}\:\mathrm{coefficient}\:\mathrm{is} \\ $$$$\lambda\:=\mathrm{C}_{\mathrm{16}} ^{\mathrm{7}} \:\:=\frac{\mathrm{16}!}{\mathrm{7}!\:\mathrm{9}!}\: \\ $$

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