Binomial Coefficient Calculator

What this calculator does

The binomial coefficient, written C(n, k) and read "n choose k", is the number that turns up in two famous places at once: as the entries of Pascal's triangle, and as the coefficients when you expand a bracket like (a + b) raised to a power. This works it out for you.

Enter n and k and it returns the binomial coefficient. It runs right here in the browser. And yes, it is the very same number you would get from a combination, seen from the algebra side rather than the counting side. More on that below.

Using the calculator

1. Enter n, the upper number.
2. Enter k, the lower number.
3. Press Calculate.

It returns the binomial coefficient C(n, k). As always, n must be at least as big as k. Reset clears both boxes.

What a binomial coefficient is

A binomial coefficient is a single whole number attached to a pair, n on top and k below, usually written with the two stacked inside a tall bracket, or as C(n, k). It counts, among other things, how many ways you can choose k items from n. But its real fame comes from the two structures it builds: Pascal's triangle and the expansion of a binomial. Those are what this page is about.

The formula

The binomial coefficient is given by:

C(n, k) = n! ÷ ( k! × (n − k)! )

with n the top number, k the bottom, and the exclamation mark meaning factorial, a whole number multiplied down to 1. It is exactly the combination formula, which is the first hint that these are the same number wearing different hats.

Pascal's triangle

Write 1 at the top. On every row below, each number is the sum of the two just above it, and the ends are always 1. You get this:

1
1  1
1  2  1
1  3  3  1
1  4  6  4  1
1  5  10  10  5  1

Every single number in that triangle is a binomial coefficient. Counting rows and positions from 0, the entry in row n at position k is exactly C(n, k). So the 10 sitting in the bottom row is C(5, 2), and its neighbour is C(5, 3), also 10. The rule that each number is the sum of the two above it is a real identity, C(n, k) = C(n − 1, k − 1) + C(n − 1, k), known as Pascal's rule. It is often the quickest way to build the coefficients without any factorials at all.

Where the name comes from: the binomial theorem

A binomial is just a two-term expression like (a + b). Raise it to a power and multiply it all out, and the numbers in front of each term are the binomial coefficients. For example:

(a + b)³ = a³ + 3a²b + 3ab² + b³

The coefficients there are 1, 3, 3, 1, which is exactly row 3 of Pascal's triangle. In general, the binomial theorem says (a + b)ⁿ adds up terms of the form C(n, k) × a^n−k × b^k, with k running from 0 to n. So if you ever need the coefficient of a particular term in an expansion, you need a binomial coefficient, and this is where the "binomial" in the name comes from.

A worked example, three ways

Take C(5, 2). Here are three routes to it, all landing on the same number.

1. By the formula: 5! ÷ (2! × 3!) = 120 ÷ (2 × 6) = 120 ÷ 12 = 10.
2. From Pascal's triangle: go to row 5 and count to position 2, and there is the 10.
3. As a coefficient: it is the number in front of a³b² when you expand (a + b)⁵, which is 10.

Three different questions, one answer. That is the nature of the binomial coefficient.

Why it is the same as "n choose k"

It is fair to ask why the coefficient in an algebra expansion should equal the number of ways to choose a group. Here is the link. When you expand (a + b)⁵, every term is built by picking either an a or a b from each of the five brackets. To land on the term a³b², you must pick b from exactly 2 of the 5 brackets. The number of ways to choose which 2 brackets give the b is C(5, 2), and that is precisely how many times the term appears, so that is its coefficient. Counting and algebra meet in the same number, which is why this tool and the combination calculator return the same result.

A note on large numbers

The tool works the coefficient out from the factorials directly, which is exact and instant for ordinary values of n. Since factorials grow enormously, past roughly n = 18 the numbers exceed the range a browser keeps perfectly exact, so for very large n the final digits may not be reliable. For the sizes you meet in most algebra and probability, it is exact. For very large n, a huge result is best read as very close rather than exact to the last digit. Pascal's rule, adding the two numbers above, sidesteps the factorials entirely and stays exact for longer if you ever need it.

Questions people ask

What is a binomial coefficient?

The number C(n, k), read "n choose k". It is the entry in Pascal's triangle at row n, position k, and the coefficient of a^n−kb^k when (a + b)ⁿ is expanded.

Is it the same as a combination?

Yes. C(n, k) is both the count of ways to choose k items from n and the binomial coefficient. Same formula, same number, two ways of looking at it.

How does it relate to Pascal's triangle?

Every number in Pascal's triangle is a binomial coefficient. The entry in row n at position k is C(n, k), and each is the sum of the two directly above it.

Why does C(5, 2) equal C(5, 3)?

Because C(n, k) = C(n, n − k), which is why Pascal's triangle reads the same forwards and backwards along each row. Both here are 10.

What is the binomial theorem?

The rule that (a + b)ⁿ expands into a sum of terms C(n, k) × a^n−k × b^k. The binomial coefficients are its coefficients.

References

A note on where this comes from. The binomial coefficient is the same quantity as the combination, and it is the standard object appearing in the binomial theorem and in Pascal's triangle. The triangle is named after Blaise Pascal, whose Traité du triangle arithmétique (1654) studied it in depth, though the pattern was known centuries earlier to mathematicians in China, Persia and India. For further reading, see Binomial coefficient.

1. Blaise Pascal, Traité du triangle arithmétique (1654), the influential study of the arithmetical triangle now named after him.
2. Weisstein, Eric W. "Binomial Coefficient", Wolfram MathWorld, on the coefficient, Pascal's rule, and the binomial theorem.
3. Graham, Knuth and Patashnik, Concrete Mathematics, a standard reference on binomial coefficients and their identities.