Question

The constant term in the expansion of ${(1+x)}^{10}$.${(1+\frac{1}{x})}^{12}$ is

• A. ${}^{22}{\text{C}}_{10}$
• B. $0$
• C. ${}^{22}{\text{C}}_{11}$
• D. none of these

Answer 1

Answer: (A)

${}^{22}{\text{C}}_{10}$

$(1+x{)}^{10}(\frac{1}{x}+1{)}^{12}$
$=\frac{1}{x^{(}12)}(1+x{)}^{22}$
$T_{r+1}={}^{22}{C}_r{x}^{r-12}$
Hence the term independent of $x$ will be the constant term.
$r-12=0$
$\Rightarrow r=12$
Hence the ${13}^{th}$ term from the starting will be the constant term.
Therefore
The corresponding binomial coefficient
$={}^{22}{C}_{12}$
$={}^{22}{C}_{22-12}$
$={}^{22}{C}_{10}$