Binomial Formula Calculator
Expand (x + y)n, view coefficients, and optionally evaluate a numeric value.
What Is the Binomial Formula?
The binomial formula (also called the binomial theorem) tells you how to expand a power of a sum:
(a + b)n = Σ C(n, k) an-kbk, for k = 0 to n.
Instead of multiplying (a + b) by itself repeatedly, this formula gives every term directly with the correct coefficient.
Why This Calculator Is Useful
- Quickly expands expressions like (x + y)12.
- Shows binomial coefficients clearly.
- Finds a specific term without writing the full expansion by hand.
- Evaluates a numeric result when values are provided.
How the Calculator Works
1) Coefficients from Combinations
Each coefficient comes from the combination formula:
C(n, k) = n! / (k!(n-k)!).
These values are the same numbers you see in Pascal’s Triangle.
2) Power Pattern
In each term, the first symbol’s exponent decreases from n to 0, while the second symbol’s exponent increases from 0 to n.
3) Specific Term Formula
The r-th term (counting from 1) of (x+y)n is:
Tr = C(n, r-1) xn-r+1 yr-1.
Example
For (x + y)4, the coefficients are 1, 4, 6, 4, 1, so:
x4 + 4x3y + 6x2y2 + 4xy3 + y4.
Common Mistakes to Avoid
- Using a negative or non-integer power for this standard form.
- Forgetting that term count is n + 1.
- Mixing up the exponents (one goes down, the other goes up).
- Using incorrect coefficient values from combinations.
Final Tip
If you are studying algebra, probability, or calculus, mastering binomial expansion saves time and prevents sign/coefficient errors. Use the calculator for speed, then verify a few smaller powers manually to build confidence.