Math & Statistics · 二项式定理（Binomial Theorem） · Updated 2026-10-25
What Is Binomial Theorem?
Definition
(a+b)^n = Σ C(n,k)·a^(n−k)·b^k with binomial coefficients C(n,k)=n!/(k!(n−k)!), forming Pascal's triangle (each entry the sum of the two above). Each term counts the ways "exactly k of n independent trials produce b" — the origin of the binomial distribution P(X=k)=C(n,k)p^k(1−p)^(n−k). For example, exactly 5 heads in 10 flips: C(10,5)/2^10 ≈ 24.6%.
Formula
(a+b)^n = Σ C(n,k)·a^(n−k)·b^k
Worked Example
(a+b)^4 = a⁴+4a³b+6a²b²+4ab³+b⁴ — coefficients 1,4,6,4,1 are row 4 of Pascal's triangle; the chance of two boys and two girls in four births is 6/16 = 37.5% (the middle coefficient).
Common Misconception
Computing 2^10 as 2×10=20 — exponentiation is repeated multiplication, not addition. Another trap: thinking "5 heads in 10 is most likely, so it must exceed 50%" — the single outcome is 24.6%; crossing 50% requires cumulative events like "at least 5 heads."
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