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The coefficient of x^r(0 \leq r \leq n-1) in the expansion : (x+2)^{n-1}+(x+2)^{n-2} \cdot(x+1)+(x+2)^{n-3} \cdot(x+1)^2+\ldots \ldots+(x+1)^{n-1} is:

Option: 1

{ }^n C_r\left(2^r-1\right)


Option: 2

{ }^n C_r\left(2^{n-r}-1\right)


Option: 3

{ }^n C_r\left(2^r+1\right)


Option: 4

{ }^n C_r\left(2^{n-r}+1\right)


Answers (1)

best_answer

\begin{aligned} E & =(x+2)^{n-1}\left[1+\frac{x+1}{x+2}+\left(\frac{x+1}{x+2}\right)^2+\ldots \ldots+\left(\frac{x+1}{x+2}\right)^{n-1}\right] \\ & =(x+2)^{n-1}\left[\frac{1-\left(\frac{x+1}{x+2}\right)^n}{1-\frac{x+1}{x+2}}\right]=(x+2)^n\left[\frac{(x+2)^n-(x+1)^n}{(x+2)^n}\right]=(x+2)^n-(x+1)^n \end{aligned}

Now co-efficient of x^r in \left[(2+x)^n-(1+x)^n\right]={ }^n C_r 2^{n-r}-{ }^n C_r={ }^n C_r\left(2^{n-r}-1\right)

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Ritika Harsh

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