Question

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

• A. $9$
• B. $0$
• C. $5$
• D. $10$

Answer 1

The correct option is C $5$

expansion of $(1+x{)}^n$

$=1{+}^n{C}_{1}x{+}^n{C}_2{x}^2+.......{+}^n{C}_n{x}^n$

$(1-x{)}^n$

$=1{-}^n{C}_{1}x{+}^n{C}_2{x}^2{-}^n{C}_3{x}^2+........$

Adding,

$(1+x{)}^n+(1-x{)}^n=2(1{+}^n{C}_2{x}^2{+}^n{C}_4{x}^4+......)$

$(1+3\sqrt{2}x{)}^9+(1-3\sqrt{2}x{)}^9$

$=2\left(1{+}^9{C}_2\right(3\sqrt{2}x{)}^2{+}^9{C}_4\left(3\sqrt{2}x{)}^4{+}^9{C}_6\right(3\sqrt{2}x{)}^6{+}^9{C}_8\left(3\sqrt{2}x{)}^8\right)$

So, there are 5 non-zero terms in the expression.