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