Question

If the coefficient of x¹⁰ in the binomial expansion of `(sqrt(x)/5^(1/4) + sqrt(5)/x^(1/3))^60` is 5^kl, where l, k ∈ N and l is coprime to 5, then k is equal to ______.

Options

• 2

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• 5

Answer 1

If the coefficient of x¹⁰ in the binomial expansion of `(sqrt(x)/5^(1/4) + sqrt(5)/x^(1/3))^60` is 5^kl, where l, k ∈ N and l is coprime to 5, then k is equal to 5.

Explanation:

Given binomial expansion is `(sqrt(x)/5^(1/4) + sqrt(5)/x^(1/3))^60`

Take general term ofbinomial expansion

T_r+1 = `""^60C_r(x^(1/2)/5^(1/4))^(60-r) (5^(1/2)/x^(1/3))r`

= `""^60C_r 5(3r - 60)/4.x(180 - 5r)/6`

The power of x should be equal to 10

`(180 - 5r)/6` = 10

⇒ r = 24

Coefficient of x¹⁰ = ⁶⁰C₂₄5³

= `underline(|60)/(underline(|24)  underline(|36)) 5^3` 

Powers of 5 in = ⁶⁰C₂₄.5³

= `5^14/(5^4 xx 5^8) xx 5^3`

= 5⁵