Binomial Theorem

Compact study note.

Summary

The binomial theorem expands powers of a sum and supplies the combinatorial coefficients used in the binomial distribution.^[1]

Prerequisites

Prerequisites: None

Notation and Assumptions

For integer $n \geq 0$ , $(\binom{n}{k}) = n! / [k! (n - k)!]$ counts $k$ -subsets of an $n$ -element set.

Essential Result

Expansion formula:

(a + b)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) a^{n - k} b^{k} .

Small Example

$(x + y)^{3} = x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}$ .

Common Mistakes

Using the theorem for noninteger exponents without the infinite-series version.
Forgetting that $k$ runs from $0$ to $n$ .

Connections

References

OpenStax, Introductory Statistics 2e, "Chapter 4: Discrete Random Variables", https://openstax.org/books/introductory-statistics-2e/pages/4-introduction ↩︎