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Coefficients of  x^r[0 \leq r \leq(n-1)]  in the expansion of (x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots+(x+2)^{n-1}

 

Option: 1

{ }^n C_r\left(3^r-2^n\right)


Option: 2

{ }^n C_r\left(3^{n-r}-2^{n-r}\right)


Option: 3

{ }^n C_r\left(3^r+2^{n-r}\right)


Option: 4

None of these


Answers (1)

best_answer

We have \begin{aligned} (x+3)^{n-1}+ & (x+3)^{n-2}(x+2)+ \ (x+3)^{n-3}(x+2)^2+\ldots+(x+2)^{n-1} \end{aligned}

=\frac{(x+3)^n-(x+2)^n}{(x+3)-(x+2)}=(x+3)^n-(x+2)^n

\left(\frac{x^n-a^n}{x-a}=x^{n-1}+x^{n-2} a^1+x^{n-3} a^2+\ldots+a^{n-1}\right)

Therefore coefficient of x^r  in the given expression

= Coefficient of x^r  in \left[(x+3)^n-(x+2)^n\right]

={ }^n C_r 3^{n-r}-{ }^n C_r 2^{n-r}={ }^n C_r\left(3^{n-r}-2^{n-r}\right)

Posted by

Ritika Jonwal

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