Find the coefficient of x 4 in the expansion of (1 + x) n (1 – x) n . Deduce that C 2 = C 0 C 4 – C 1 C 3 + C 2 C 2 – C 3 C 1 + C 4 C 0 , where C r stands for n C r .

Question

Philoid · Accepted Answer

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

x⁰.x⁴ = C₀C₄

x¹.x³ = - C₁C₃

x².x² = C₂C₂

x³.x¹= - C₃C₁

x⁴.x⁰ = C₄C₀

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,

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

= (1)(1) – (4)(4) + (6)(6) – (4)(4) + (1)(1)

= 1 – 16 + 36 – 16 + 1

= 6

= ⁴C₂

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

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

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.