Show that the ratio of the coefficient of x¹⁰ in the expansion of (1 – x2)¹⁰ and the term independent of x in the expansion of is 1 : 32.

To Prove : coefficient of x¹⁰ in (1-x²)¹⁰: coefficient of x⁰ in = 1:32

For (1-x²)¹⁰ ,

Here, a=1, b=-x² and n=15

We have formula,

To get coefficient of x¹⁰ we must have,

(x)^2r = x¹⁰

• 2r = 10

• r = 5

Therefore, coefficient of x¹⁰

For ,

Here, a=x, and n=10

We have a formula,

Now, to get coefficient of term independent of xthat is coefficient of x⁰ we must have,

(x)^10-2r = x⁰

• 10 - 2r = 0

• 2r = 10

• r = 5

Therefore, coefficient of x⁰

Therefore,

Hence,

coefficient of x¹⁰ in (1-x²)¹⁰: coefficient of x⁰ in = 1:32