Revise: The Binomial Theorem
Pascal's triangle and binomial coefficients, expanding small binomial powers, and the general binomial theorem for finding any term or coefficient directly.
Meaning of Binomial Coefficients and Pascal's Triangle
ⁿCᵣ = n!/(r!(n−r)!) — ways to choose r from n.
⁵C₂ = 10.
Finding the Coefficient of a Specific Term
Set the exponent of x equal to target, solve for r.
Coeff of x⁴ in (x+3)⁶ → 135.
Meaning of Binomial Coefficients and Pascal's Triangle
ⁿCᵣ counts combinations; also the coefficients of (a+b)ⁿ.
⁶C₂ = 15.
The Binomial Theorem
General formula for any n — no triangle needed.
Coefficient of x³ in (x+2)⁴ = ⁴C₁×2 = 8.
Finding the Coefficient of a Specific Term
Solve n−r = target power for r, then evaluate ⁿCᵣ×bʳ.
Coeff of x³ in (x+2)⁵ → 40.