Math

Binomial Expansion Calculator

Expand Any Binomial Power

Enter n to see the binomial coefficients (the n-th row of Pascal's triangle) that appear in the expansion of (a+b)^n. The calculator shows each coefficient C(n,k) and the full Pascal's triangle up to row n, making it easy to understand the relationship between the triangle and polynomial expansion.

Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Lottery Draw

Calculate the number of possible 6/49 lottery combinations.

Key values: Total balls: 49 · Drawn: 6 · Combination mode

Race Podium

Find the number of ways to award Gold, Silver, and Bronze among 8 runners.

Key values: Runners: 8 · Medals: 3 · Permutation mode

Secret Santa

Calculate valid gift assignments where nobody draws their own name.

Key values: Participants: 8 · Derangement mode

Password Keyspace

Estimate the number of possible 8-character passwords from printable ASCII.

Key values: Character set: 94 · Length: 8 · Repetition allowed

Documentation

This calculator is also known as Binomial Expansion Calculator.

Read the complete guide

The Binomial Theorem

(a+b)^n expands into a sum of terms where each coefficient is a binomial coefficient C(n,k). Row n of Pascal's triangle gives all the coefficients at once.

Examples

Expanding (a+b)^4

Find the coefficients of (a+b)^4.

Row 4 of Pascal's triangle is [1, 4, 6, 4, 1], so (a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.

Key takeaway: The coefficients are symmetric: C(4,0) = C(4,4) = 1, C(4,1) = C(4,3) = 4.

Working with Binomial Expansions

Tips for expanding binomial powers:

Use Pascal's triangle to read off all coefficients of (a+b)^n at once
The sum of row n equals 2^n (set a=b=1 in the expansion)
The alternating sum of row n equals 0 (set a=1, b=-1) for n >= 1

Frequently Asked Questions about Binomial Expansion Calculator

How does Pascal's triangle relate to binomial expansion?

Row n of Pascal's triangle contains the coefficients for expanding (a+b)^n. For example, row 3 is [1, 3, 3, 1], so (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.

What is the general term in a binomial expansion?

The k-th term (k starting at 0) of (a+b)^n is C(n,k) × a^(n-k) × b^k. For example, the second term (k=1) of (a+b)^5 is C(5,1) × a⁴ × b = 5a⁴b.

Can this be used for (a−b)^n or (1+x)^n?

Yes. For (a−b)^n, substitute b with −b so odd-indexed terms become negative. For (1+x)^n, set a = 1. The binomial theorem also generalizes to non-integer and negative exponents via the generalized binomial series.