Binomial Expansions Notes — Sec 3 A-Math

Binomial coefficient

$\binom{n}{r} = \frac{n!}{r!(n-r)!}, \quad 0 \le r \le n$

Cancel factorials rather than computing them in full. Endpoint values: ⁿC₀ = 1, ⁿC₁ = n, ⁿCₙ₋₁ = n, ⁿCₙ = 1.

Symmetry of binomial coefficients

$\binom{n}{r} = \binom{n}{n-r}$

Choosing r objects to take is the same as choosing n − r to leave. The list ⁿC₀, ⁿC₁, …, ⁿCₙ reads the same forwards and backwards, which makes it a check.

Binomial theorem (positive integer n)

$(a+b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{r}a^{n-r}b^r + \cdots + \binom{n}{n}b^n$

There are exactly n + 1 terms. Powers of a descend from n to 0, powers of b ascend from 0 to n, and in every term they add up to n.

General term of (a + b)ⁿ

$T_{r+1} = \binom{n}{r} a^{n-r} b^{r}$

a and b are the entire terms, signs and coefficients included. For the k-th term, use r = k − 1.

Coefficient of a product

$\text{coeff of } x^k \text{ in } (a+bx)f(x) = a c_k + b c_{k-1}$

cᵣ is the coefficient of xʳ in the expansion of f(x). The bx multiplier supplies one power of x itself, so it pairs with the xᵏ⁻¹ term.

Binomial coefficients in terms of n

$\binom{n}{1} = n, \quad \binom{n}{2} = \frac{n(n-1)}{2}, \quad \binom{n}{3} = \frac{n(n-1)(n-2)}{6}$

Use these when the index n is unknown. Include the numerical power from the bracket, e.g. ⁿC₂ × 4² in (1 + 4x)ⁿ.