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Question Number 103826 by bobhans last updated on 17/Jul/20
In the expansion of (1+x)^(20)  if the  coefficient of x^r  is twice the coefficient  of x^(r−1) , what the value of the  coefficient?
$${In}\:{the}\:{expansion}\:{of}\:\left(\mathrm{1}+{x}\right)^{\mathrm{20}} \:{if}\:{the} \\ $$$${coefficient}\:{of}\:{x}^{{r}} \:{is}\:{twice}\:{the}\:{coefficient} \\ $$$${of}\:{x}^{{r}−\mathrm{1}} ,\:{what}\:{the}\:{value}\:{of}\:{the} \\ $$$${coefficient}?\: \\ $$
Answered by bramlex last updated on 17/Jul/20
(1+x)^(20)  = Σ_(r = 0) ^(20) C _r^(20)  1^r .x^(20−r)   coefficient of x^r  is  (((20)),((  r)) ) and coefficient of x^(r−1)  is  (((   20)),(( r−1)) )   so condition in equation    (((20)),((  r)) ) = 2 (((    20)),((  r−1)) )   note  ((n),(r) ) = ((n−r+1)/r)  (((  n)),((r−1)) )  so your equation reduces to  ((20−r+1)/r) = 2 ⇒ r = 7. therefore  the value of coefficient ((20!)/(7!.13!))  = ((20.19.18.17.16.15.14)/(7.6.5.4.3.2.1))
$$\left(\mathrm{1}+{x}\right)^{\mathrm{20}} \:=\:\underset{{r}\:=\:\mathrm{0}} {\overset{\mathrm{20}} {\sum}}{C}\:_{{r}} ^{\mathrm{20}} \:\mathrm{1}^{{r}} .{x}^{\mathrm{20}−{r}} \\ $$$${coefficient}\:{of}\:{x}^{{r}} \:{is}\:\begin{pmatrix}{\mathrm{20}}\\{\:\:{r}}\end{pmatrix}\:{and}\:{coefficient}\:{of}\:{x}^{{r}−\mathrm{1}} \:{is}\:\begin{pmatrix}{\:\:\:\mathrm{20}}\\{\:{r}−\mathrm{1}}\end{pmatrix}\: \\ $$$${so}\:{condition}\:{in}\:{equation}\: \\ $$$$\begin{pmatrix}{\mathrm{20}}\\{\:\:{r}}\end{pmatrix}\:=\:\mathrm{2}\begin{pmatrix}{\:\:\:\:\mathrm{20}}\\{\:\:{r}−\mathrm{1}}\end{pmatrix}\: \\ $$$${note}\:\begin{pmatrix}{{n}}\\{{r}}\end{pmatrix}\:=\:\frac{{n}−{r}+\mathrm{1}}{{r}}\:\begin{pmatrix}{\:\:{n}}\\{{r}−\mathrm{1}}\end{pmatrix} \\ $$$${so}\:{your}\:{equation}\:{reduces}\:{to} \\ $$$$\frac{\mathrm{20}−{r}+\mathrm{1}}{{r}}\:=\:\mathrm{2}\:\Rightarrow\:{r}\:=\:\mathrm{7}.\:{therefore} \\ $$$${the}\:{value}\:{of}\:{coefficient}\:\frac{\mathrm{20}!}{\mathrm{7}!.\mathrm{13}!} \\ $$$$=\:\frac{\mathrm{20}.\mathrm{19}.\mathrm{18}.\mathrm{17}.\mathrm{16}.\mathrm{15}.\mathrm{14}}{\mathrm{7}.\mathrm{6}.\mathrm{5}.\mathrm{4}.\mathrm{3}.\mathrm{2}.\mathrm{1}}\:\: \\ $$

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