Expansion of (a+b)ⁿ into n+1 terms using binomial coefficients. Pascal's triangle gives the coefficients.
Probability calculations — Binomial distribution P(X=k) = C(n,k)p^k(1-p)^(n-k).
Financial options — Binomial model for option pricing uses powers of up/down factors.
Genetics — Mendel's pea experiments: probability of traits in offspring.
Combinatorics — Number of ways to select items: C(n,k) = n!/(k!(n-k)!).
Error correction — Hamming codes use binomial coefficients for redundancy.
Pascal's triangle — Each coefficient is sum of two above it.
Approximations — (1+x)ⁿ ≈ 1+nx for small x; used in physics and engineering.
Interest calculations — (1+r)ⁿ expansion for compound interest formulas.
Polling predictions — Margin of error uses binomial confidence intervals.
Sports statistics — Probability of exactly k wins in n games.
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