Question

Find the coefficient of x⁴ in the expansion `(1 + x^3)^50 (x^2 + 1/x)^5`

Answer 1

`(x^2 + 1/x)^5 = ""^5"C"_0 (x^2)^5 + ""^5"C"_1 (x^2)^4 (1/x) + ""^5"C"_2 (x2)^3 (1/x)^2 + ""^5"C"_3 (x^2)^2 (1/x)^3 + ""^5"C"_4 (x^2) (1/x)^4 + ""^5"C"_5 (1/x)^5`

⁵C₀ = 1 = ⁵C₅ ; ⁵C₁ = 5 = ⁵C₄ ; ⁵C₂ = `(5 xx 4)/(2 xx 1)` = 10 = ⁵C₃

= `x^10 + 5(x^8) (1/x) + 10(x^6) (1/x^2) + 10(x^4) (1/x^3)+ 5(x^2) (1/x^4) + 1/x^5` 

= `x^10 + 5x^7 + 10x^4 + 10x + 5/x^2 + 1/x^5`

(1 + x³)⁵⁰ = `""^50"C"_0 (1)^50 (x^3)^0 + ""^50"C"_1 (1)^49 (x^3)^1 + ""^50"C"_2 (1)^48 (x^3)^2 +""^50"C"_3 (1)^7 (x^3)^3 + .... ""^50"C"_50 (1)^circ(x^3)^50`

⁵⁰C₀ = 1 = ⁵⁰C₅₀, ⁵⁰C₁ = 50, ⁵⁰C₂ = 1225, ⁵⁰C₃ = 19600

= (50) + (50)x³ + 1225x⁶ 7600x⁹.... x¹⁵⁰

To find coefficient of x⁴

`(1 + x^3)^50 (x^2 + 1/x)^5 = (1 + 50x^3 + 1225x^6 + 19600x^9 ...  x^150) xx (x^10 + 5x^7 + 10x^4 + 10x^4 + 10x + 5/x^2 + 1/x^5)`

When multiplying these terms, we get x⁴ terms

= `(1 xx 10x^4) + (50x^3 xx 10x) + (1225x^6 xx5/x^2) + (19600x^9 xx 1/x^5)`

= 10x⁴ + 500x⁴ + 6125x⁴ + 19600x⁴

= 26325x⁴

∴ The coefficient of x⁴ is 26325