What this calculator does
The binomial theorem expands a bracket raised to a power without multiplying it out repeatedly. The coefficients are exactly the rows of Pascal's triangle, which is why the two topics are always taught together.
The coefficients always sum to two to the power. For the fifth power they run 1, 5, 10, 10, 5, 1 and total 32, which is the value of the expansion when both terms are 1.
The formula
Each coefficient is a binomial coefficient, the number of ways to choose k items from n. The expansion runs from the full power of the first term down to the full power of the second.
| Term | Meaning |
|---|---|
| Binomial coefficient | C(n,k), the number of ways to choose k from n. |
| Pascal triangle | The array of binomial coefficients, each the sum of the two above it. |
| Symmetry | The coefficients read the same forwards and backwards. |
The inputs explained
| Field | What to enter |
|---|---|
| Power (n) | The power to raise the binomial to. Accepts 0 to 30. |
| Value of x | The value of the first term, used for the numerical check. |
| Value of y | The value of the second term, used for the numerical check. |
When to use it
Expanding a bracket
The theorem avoids multiplying the bracket out term by term.
Finding one specific term
A single binomial coefficient gives any term without expanding the rest.
Probability calculations
The binomial distribution uses exactly these coefficients.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What are the coefficients at each power?
Four powers and their expansions.
| Power | Coefficients (row of Pascal triangle) | Sum of all coefficients (2ⁿ) |
|---|---|---|
| n = 0 | 1 | 1 |
| n = 3 | 1, 3, 3, 1 | 8 |
| n = 5 | 1, 5, 10, 10, 5, 1 | 32 |
| n = 8 | 1, 8, 28, 56, 70, 56, 28, 8, 1 | 256 |
Questions
Why do the coefficients sum to 2 to the power?
Because setting both terms to 1 makes the expansion equal 2 raised to the power, while every term reduces to just its coefficient. The sum of the coefficients therefore has to equal that value.
Why are the coefficients symmetric?
Because choosing k items from n is the same as choosing which n minus k to leave out. That symmetry in the counting produces the symmetry you see in every row of Pascal's triangle.
How do I find just one term?
Use the general term formula directly: the term with y to the power k is C(n,k) times x to the power n minus k times y to the power k. There is no need to expand the whole thing.
Does it work for negative or fractional powers?
Yes, but the expansion becomes an infinite series rather than a finite sum, and it only converges under certain conditions. This calculator handles the whole-number case.
For the triangle itself, see the Pascal triangle calculator. For binomial probability, see the binomial probability calculator.