Question

The number of non zero terms in the expansion of $(1+3\sqrt{2}x{)}^9+(1-3\sqrt{2}x{)}^9$ is

• A. $9$
• B. $0$
• C. $5$
• D. $10$

Answer 1

The correct option is C $5$

$(1+3\sqrt{2}x){}^9+(1-3\sqrt{2}x){}^9$
Applying binomial theorem for expansion of the following
$=1+\left(3\sqrt{2}x\right){(}^9{C}_1)+(3\sqrt{2}x{)}^2{(}^9{C}_2)...({}^9{C}_9\left(3\sqrt{2}x{)}^9\right)$
$+1-\left(3\sqrt{2}x\right){(}^9{C}_1)+(3\sqrt{2}x{)}^2{(}^9{C}_2)...-({}^9{C}_9\left(3\sqrt{2}x{)}^9\right)$
$=2[1+(3\sqrt{2}x\left){}^2{(}^9{C}_2\right)+\left(3\sqrt{2}x\right){}^4{(}^9{C}_4)...(3\sqrt{2}x\left){}^8\right({}^9{C}_8\left)\right]$
The powers of $3\sqrt{2}x$ are in A.P
$0,2,4,6,8$
Hence, including the term $1or{x}^0$ there are $\frac{9+1}{2}$ terms.
Hence, there are $5$ terms,