Binomial Theorem — Mathematics Knowledge System

Overview

The Binomial Theorem gives a direct formula for expanding (x+y)ⁿ into a sum of terms, without multiplying out (x+y) by itself n times. Each term's coefficient is a binomial coefficient — the same numbers that appear in Pascal's Triangle and count combinations — making the theorem a bridge between algebra and combinatorics.

Formal Definition

Definition

For any nonnegative integer n, the binomial (x+y)ⁿ expands as:

(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Binomial Theorem

\binom{n}{k} = \frac{n!}{k!(n-k)!}

The binomial coefficient, the number of ways to choose k items from n

Properties

Symmetry

\binom{n}{k} = \binom{n}{n-k}

Number of terms

(x+y)^n \text{ expands to exactly } n+1 \text{ terms}

Pascal's rule

\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}

Sum of coefficients

\sum_{k=0}^{n}\binom{n}{k} = 2^n

Worked Examples

Use row 4 of Pascal's Triangle for the coefficients: 1, 4, 6, 4, 1.
$(x+y)^4 = \binom{4}{0}x^4 + \binom{4}{1}x^3y + \binom{4}{2}x^2y^2 + \binom{4}{3}xy^3 + \binom{4}{4}y^4$
Evaluate each binomial coefficient: C(4,0)=1, C(4,1)=4, C(4,2)=6, C(4,3)=4, C(4,4)=1.
$= x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

Answer: (x+y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

Practice Problems

Difficulty 5/10

Find the coefficient of x³y² in the expansion of (x+y)⁵.

Common Mistakes

Common Mistake

Forgetting the alternating signs when expanding (x-y)ⁿ.

Rewrite (x-y)ⁿ as (x+(-y))ⁿ before applying the theorem: each term becomes C(n,k)xⁿ⁻ᵏ(-y)ᵏ, so odd-k terms carry a negative sign.

Summary

The Binomial Theorem: (x+y)ⁿ = Σ C(n,k) xⁿ⁻ᵏyᵏ for k=0 to n.
Coefficients C(n,k) = n!/(k!(n-k)!) are exactly the entries of row n of Pascal's Triangle.
The expansion always has n+1 terms, and the coefficients are symmetric: C(n,k)=C(n,n-k).
For (x-y)ⁿ, rewrite as (x+(-y))ⁿ so signs alternate correctly term by term.

References

WebsiteWikipedia — Binomial theorem