What is the binomial theorem?

The binomial theorem states that (a + b) to the power n equals the sum from k=0 to n of C(n,k) times a to the power (n minus k) times b to the power k, where C(n,k) is the binomial coefficient n choose k, equal to n factorial divided by k factorial times (n minus k) factorial. These coefficients appear as rows of Pascal's triangle.

What is (x+y)^4 expanded?

Using the binomial theorem with n=4: (x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4. The coefficients 1, 4, 6, 4, 1 are the fifth row of Pascal's triangle (row index 4). Each term has total degree 4 (the exponents of x and y sum to 4 in every term).

How do Pascal's triangle and the binomial theorem relate?

Each row of Pascal's triangle (starting from row 0 at the top) gives the binomial coefficients for expanding (a+b) to the corresponding power. Row 4 is 1, 4, 6, 4, 1, which are the coefficients of (a+b)^4. Each entry is the sum of the two entries directly above it in the previous row.

Binomial Expansion Calculator

The binomial theorem gives a formula for expanding any expression of the form (a + b) raised to a positive integer power n. Instead of multiplying (a + b) by itself n times, the theorem lets you write down the result directly using a set of coefficients called binomial coefficients, written C(n,k) or n-choose-k, which count the number of ways to choose k items from n. The expanded form has n+1 terms: the first is a to the power n, the last is b to the power n, and each intermediate term has a multiplying coefficient that follows the pattern of a row in Pascal's triangle. Pascal's triangle is a triangular array where each entry is the sum of the two above it; its rows are exactly the binomial coefficients. The binomial theorem appears throughout algebra (factoring and manipulating expressions), probability (the binomial distribution), calculus (Taylor series approximations), and combinatorics. This calculator expands (x + y) to the power n for any n from 0 to 10. You can substitute any terms for x and y, including negative values, which handles expressions like (a - b)^n by treating b as negative. The output shows the symbolic expansion, each term with its coefficient, and the relevant Pascal's triangle row. The default is (x+y)^4 which gives x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4.

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Power (n, 0 to 10)

First term label (a)

Second term label (b)

x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

(x+y)^4 expansion

Terms5

Sum of coefficients16

Pascal row 4: 1 4 6 4 1

Exponent labels use standard superscript notation. Substitute any symbols for a and b.

How it works

The binomial coefficient C(n,k) is computed as n! / (k! x (n-k)!). For each k from 0 to n, the term is C(n,k) times a^(n-k) times b^k. The symbolic expansion labels powers with superscript HTML entities (squared, cubed, etc.) and omits coefficient 1 and exponent 1 for readability. Pascal's triangle row n is the array of coefficients C(n,0) through C(n,n). The sum of coefficients equals 2^n, which you get by setting a=b=1.

Worked example

(x+y)^4 with n=4. Pascal's triangle row 4 is 1, 4, 6, 4, 1. The five terms are: C(4,0)x^4y^0 = x^4; C(4,1)x^3y = 4x^3y; C(4,2)x^2y^2 = 6x^2y^2; C(4,3)xy^3 = 4xy^3; C(4,4)y^4 = y^4. Full expansion: x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4. Sum of coefficients: 1+4+6+4+1 = 16 = 2^4. These match the defaults pre-filled above.

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