Question

The number of terms in the expansion of $(x+1{)}^{2n+1}-(x-1{)}^{2n+1},$ is

• A. $2n$
• B. $n$
• C. $n+1$
• D. $2n+1$

Answer 1

The correct option is C $n+1$

EXpansion is ${(x+1)}^{2n+1}-{(x-1)}^{2n+1}$

Number of terms in each expansion

$=2n+2$

Expansion $=[C_0{x}^{2n+1}+C_1{x}^{2n}+C_2{x}^{2n-1}+C_3{x}^{2n-2}+C_4+.....+C_{2n+1}]-[C_0{x}^{2n+1}-C_1{x}^{2n}+C_2{x}^{2n-1}-C_3{x}^{2n-2}+...]$

$=C_0{x}^{2n+1}+C_1{x}^{2n}+C_2{x}^{2n-1}+C_3{x}^{2n-2}+.....-C_0{x}^{2n+1}+C_1{x}^{2n}-C_2{x}^{2n-1}+C_3{x}^{2n-2}+.......$

$=2[C_1{x}^{2n}+C_3{x}^{2n-2}+C_5{x}^{2n-4}+.....]$

$\Rightarrow$Half the terms gets cancelled

$\Rightarrow$other half terms $=\frac{2n+2}{2}$

$=n+1$

$\Rightarrow$ Number of terms $=n+1$