Question

the constant term in the expansion of ${(1+x+\frac{2}{x})}^6$

• A. $581$
• B. $569$
• C. $517$
• D. $479$

Answer 1

The correct option is A $581$

$\begin{array}{cc}& {(1+x+\frac{2}{x})}^6\\ & 6C_0(1+x{)}^6+{}^6\left(1\right(1+x{)}^5\left(\frac{2}{x}\right)+{}^6\left(2\right(1+x{)}^4{\left(\frac{2}{x}\right)}^2\\ +& 6C_3(1+x{)}^3{\left(\frac{2}{x}\right)}^3\end{array}$

Further terms will not be constant

${}^6{C}_0{}^6{C}_0+{}^6{C}_{1}5{}^5{C}_1\left(2\right)+{}^6{C}_2{}^4{C}_2(2{)}^2$

$+6C_3{}^3{C}_3(2{)}^3$

$1+60+4\left(15\right)6+160=581$