Binomial Theorem And Its Simple Applications Question 163
Question: Coefficients of $ x^{r}[0\le r\le (n-1)] $ in the expansion of $ {{(x+3)}^{n-1}}+{{(x+3)}^{n-2}}(x+2) $ $ +{{(x+3)}^{n-3}}{{(x+2)}^{2}}+…+{{(x+2)}^{n-1}} $
Options:
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
Show Answer
Answer:
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}}) $
BETA
