A) \[^{n}{{C}_{r}}({{3}^{r}}-{{2}^{n}})\]
B) \[^{n}{{C}_{r}}({{3}^{n-r}}-{{2}^{n-r}})\]
C) \[^{n}{{C}_{r}}({{3}^{r}}+{{2}^{n-r}})\]
D) None of these
Correct Answer: B
Solution :
We have \[{{(x+3)}^{n-1}}+{{(x+3)}^{n-2}}(x+2)+\]\[{{(x+3)}^{n-3}}{{(x+2)}^{2}}+....+{{(x+2)}^{n-1}}\] \[=\frac{{{(x+3)}^{n}}-{{(x+2)}^{n}}}{(x+3)-(x+2)}={{(x+3)}^{n}}-{{(x+2)}^{n}}\]\[(\because \frac{{{x}^{n}}-{{a}^{n}}}{x-a}={{x}^{n-1}}+{{x}^{n-2}}{{a}^{1}}+{{x}^{n-3}}{{a}^{2}}+....+{{a}^{n-1}})\] Therefore coefficient of \[{{x}^{r}}\] in the given expression = Coefficient of \[{{x}^{r}}\] in \[[{{(x+3)}^{n}}-{{(x+2)}^{n}}]\] \[={{\,}^{n}}{{C}_{r}}{{3}^{n-r}}-{{\,}^{n}}{{C}_{r}}{{2}^{n-r}}={{\,}^{n}}{{C}_{r}}({{3}^{n-r}}-{{2}^{n-r}})\]
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