The coefficient of x20 in the expansion of (1+x2)40 ⋅(x2+2+(1/x2))-5 is

Question

The coefficient of x20 in the expansion of (1+x2)40 ⋅(x2+2+(1/x2))-5 is (A)  20 C 10 (B)  30 C 25 (C) 1 (D) 0

Tardigrade Admin · Accepted Answer

∵(1+x2) 40 ⋅(x2+2+(1/x2))-5 =(1+x2)20 ⋅ x10 =(1+x2)30 ⋅ x20 ∴ Coefficient of x10 in (1+x2)30 ⋅ x10 ⇒ Coefficient of x10 in (1+x2)30 is 30 C5 or 30 C 25

Q. The coefficient of $x^{20}$ in the expansion of $(1 + x^{2})^{40} \cdot (x^{2} + 2 + \frac{1}{x ^{2}})^{- 5}$ is

$^{20} C_{10}$

$^{30} C_{25}$

1

0

$∵ (1 + x^{2})^{40} \cdot (x^{2} + 2 + \frac{1}{x ^{2}})^{- 5}$
$= (1 + x^{2})^{20} \cdot x^{10}$
$= (1 + x^{2})^{30} \cdot x^{20}$
$∴$ Coefficient of $x^{10}$ in $(1 + x^{2})^{30} \cdot x^{10}$
$\Rightarrow$ Coefficient of $x^{10}$ in $(1 + x^{2})^{30}$ is $^{30} C_{5}$ or $^{30} C_{25}$

Q. The coefficient of x20 in the expansion of (1+x2)40⋅(x2+2+x21​)−5 is

20C10​

30C25​

1

0

∵(1+x2)40⋅(x2+2+x21​)−5 =(1+x2)20⋅x10 =(1+x2)30⋅x20 ∴ Coefficient of x10 in (1+x2)30⋅x10 ⇒ Coefficient of x10 in (1+x2)30 is 30C5​ or 30C25​

Q. The coefficient of $x^{20}$ in the expansion of $(1 + x^{2})^{40} \cdot (x^{2} + 2 + \frac{1}{x ^{2}})^{- 5}$ is

$^{20} C_{10}$

$^{30} C_{25}$

$∵ (1 + x^{2})^{40} \cdot (x^{2} + 2 + \frac{1}{x ^{2}})^{- 5}$
$= (1 + x^{2})^{20} \cdot x^{10}$
$= (1 + x^{2})^{30} \cdot x^{20}$
$∴$ Coefficient of $x^{10}$ in $(1 + x^{2})^{30} \cdot x^{10}$
$\Rightarrow$ Coefficient of $x^{10}$ in $(1 + x^{2})^{30}$ is $^{30} C_{5}$ or $^{30} C_{25}$