Find the middle term in the expansion of

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.

Prove that the coefficient of xn in the binomial expansion of (1 + x)²ⁿ is twice the coefficient of xⁿ in the binomial expansion of (1 + x)^2n-1.

Show that the term independent of x in the expansion of is -252.

If the coefficients of x² and x³ in the expansion of (3 + px)⁹ are the same then prove that .