Question

Find the coefficient of x⁴ in the expansion of (1 + x)ⁿ (1 – x)ⁿ . Deduce that C₂ = C₀C₄ – C₁C₃ + C₂C₂ – C₃C₁ + C₄C₀, where C_r stands for ⁿC_r.

Answer 1

To Find : Coefficients of x⁴ 

For (1+x)ⁿ 

a =1, b = x

We have a formula,

For (1-x)ⁿ 

a = 1, b = -x and n = n 

We have formula,

Coefficients of x⁴ are

Therefore, Coefficient of x⁴ 

= C₄C₀ - C₁C₃ + C₂C₂ - C₃C₁ + C₄C₀ 

Let us assume, n=4, it becomes 

⁴C₄ ⁴C₀ - ⁴C₁ ⁴C³ + ⁴C₂ ⁴C₂ - ⁴C₃ ⁴C₁ + ⁴C₄ ⁴C₀ 

We know that,

By using above formula, we get,

Therefore, in general,

C₄C₀ - C₁C₃ + C₂C₂ - C₃C₁ + C₄C₀ = C₂ 

Therefore, Coefficient of x⁴ = C₂ 

Conclusion : 

Coefficient of x⁴ = C₂ 

C₄C₀ - C₁C₃ + C₂C₂ - C₃C₁ + C₄C₀ = C₂