# Binomial Coefficient Calculator

Welcome to the **binomial coefficient calculator**, where you'll get the chance to calculate and learn all about the mysterious `n`

choose `k`

formula. The expression denotes the number of combinations of `k`

elements there are from an `n`

-element set, and corresponds to the **nCr button on a real-life calculator**. For the answer to the question "*What is a binomial?*," the meaning of combination, the solution to "4 choose 2," and the comparison of permutation vs. combination, go ahead and **scroll down to the sections below**!

## What is a binomial?

In mathematics (algebra to be precise), **a binomial** is a polynomial with two terms (that's where the "bi-" prefix comes from). For example, the expressions `x + 1`

, `xy - 2ab`

, or `x³z - 0.5y⁵`

are all binomials, but `x⁵`

, `a + b - cd`

, or `x² - 4x²`

are not (the last one does have two terms, but we can simplify that expression to `-3x²`

, which has only one).

Now that we know what a binomial is, let's take a closer look at taking an exponent of one:

`(x² - 3)³`

.

There are some special cases of that expression - **the short multiplication formulas** you may know from school:

`(a + b)² = a² + 2ab + b²`

,

`(a - b)² = a² - 2ab + b²`

.

The polynomial that we get on the right hand side is called **the binomial expansion** of what we had in the brackets. Believe it or not, we can find their formulas **for any positive integer power**. In full generality, **the binomial theorem** tells us what this expansion looks like:

(a + b)^{n} = C_{0}a^{n} + C_{1}a^{n-1}b + C_{2}a^{n-2}b^{2} + ... + C_{n}b^{n},

where,

`Cₖ`

is the number of all possible**combinations of**`k`

**elements from an**`n`

**-element set**.

Also, for a given `n`

, these numbers are neatly presented for consecutive values of `n`

in the rows of the so-called Pascal's triangle, where a single row as a whole counts all possible subsets of the set (i.e., the cardinality of the power set).

And this marks a good moment for us to check out the meaning of "**combination**" - as we've mentioned so many times already.

## A combination: meaning

Imagine that **you're a college student**, taking a casual nap during a lecture. Suddenly, the teacher brings you back to earth by saying, "*Let's choose the groups for the mid-term projects at random.*" Well, it looks like you'll have to do some work, after all.

The problem is that **there's only one guy that you'd like to work with on the project**. If there are **twenty people in the group**, and the teacher divides you into **groups of four**, how probable is it that you'll be with your friend?

Every possible group is **an example of a combination**. In this case, a combination of four elements from a twenty-element set, or, if you prefer, of **four students from a twenty-person group**. If you'd like to get a bit technical, choosing a combination means picking a subset of a larger set. What is most important here is that **the order of the elements we choose doesn't matter**. After all, all members of a project team are equal (except those that don't do any work).

The number of combinations of `k`

elements from a set of `n`

elements is denoted by

(like a fraction of `n`

divided by `k`

but without the line in between) which we read as "**n choose k**." This is also the symbol that appears **when we choose push nCr on a calculator** (not our binomial coefficient calculator, but a regular, real-life one). For example,

is "4 choose 2" and

is "6 choose 2." In some textbooks, **the binomial coefficient is also denoted by** `C(n,k)`

, making it a function of `n`

and `k`

. "*And how do I calculate it?*" Well, easily enough. The `n`

**choose** `k`

**formula** is

`n! / (k! * (n - k)!)`

.

The exclamation mark is called a factorial. The expression `n!`

is **the product of the first** `n`

**natural numbers**, i.e.,

`n! = 1 * 2 * 3 * ... * n`

.

This means that, for example, the 4 choose 2 from above is

`4! / (2! * (4 - 2)!) = (1 * 2 * 3 * 4) / (1 * 2 * 1 * 2) = 6`

,

and 6 choose 2 is

`6! / (2! * (6 - 2)!) = (1 * 2 * 3 * 4 * 5 * 6) / (1 * 2 * 1 * 2 * 3 * 4) = 15`

.

So we can choose two elements from a set of four in six different ways, and from a set of six in fifteen ways.

Before we move on, **let's take one more look at the** `n`

**choose** `k`

**formula**. We can get from it a quite interesting **symmetric property**.

If we take `n`

choose `n - k`

, then we'll get

`n! / ((n - k)! * (n - (n - k))!) = n! / ((n - k)! * k!)`

which is the same as `n`

choose `k`

(since multiplication is commutative). In other words, we have

or `C(n,k) = C(n,n-k)`

in the other notation.

## Permutation vs. combination

In **what a factorial is**. In combinatorics, it denotes the number of permutations. **A permutation of length** `n`

means putting `n`

elements in some order. For example, if we have three cute kitten expressions, say 😹, 😻, and 🙀, then we can order them in six different ways:

(😹, 😻, 🙀)

(😹, 🙀, 😻)

(😻, 😹, 🙀)

(😻, 🙀, 😹)

(🙀, 😹, 😻)

(🙀, 😻, 😹).

Observe that this **agrees with what the factorial tells us**:

`3! = 3 * 2 * 1 = 6`

.

Note that **we can also understand this formula like this**: we choose the first element out of three (3 options), the second out of the two remaining (because we've already chosen one - 2 options), and the third out of the one that's left (because we've already chosen two - 1 option). We multiply the number of choices: `3 * 2 * 1 = 6`

, and get the factorial.

When we compare permutation vs. combination, **the keyword is order**. As we've said , the meaning behind a combination is **picking a few elements from a bigger collection**. In essence, we say which ones we pick, but not which is first, second, etc. They form a set as a whole.

A permutation, however, **puts the elements in a fixed order**, one after the other, making it a sequence rather than a set. Moreover, a permutation uses all elements from the set we've had, while a combination only chooses some of them.

As an example, once again, put yourself in the college student's shoes. When the teacher chose the group for you, **they picked a combination**. And when it comes time to present your project, and they ask one question to each of you, **they choose a permutation** (determining the order in which they ask you the questions). And we all know how important the order can be for your final grade.

## Example: using the binomial coefficient calculator

**Binomial coefficients** are one of the most important number sequences in discrete mathematics and combinatorics. They appear very often in , and are perhaps most important in the binomial distribution (the positive and the negative version). **Does that mean that only geeky mathematicians have any real use for it?**

**Not at all!** Every gambling game is based on chance, and binomial coefficients **are the vital player**. A simple coin toss is the easiest example. Let's, however, go one step further and take a look at poker.

Have you ever wondered **why some hands in poker are more valuable than others**? That's simply because **they are rarer** (unless someone's cheating, but we've seen enough gangster TV series to know that it's usually a bad idea).

There are **52 cards** in a regular deck, and in Texas hold 'em, **a player gets five cards**. Our binomial coefficient calculator and the `n`

choose `k`

formula (in our case with `n = 52`

and `k = 5`

) tells us this translates to `2,598,960`

possible hands in a game of poker. **Quite a lot**, don't you think? And now consider the best possible hand - **a royal flush in clubs** (Ace, King, Queen, Jack, and 10). This hand can happen **only in one case** - when we get exactly those cards. This means that it is **a** `1`

**in** `2,598,960`

**chance** to have it. We wouldn't recommend putting all of your savings on those odds.

Let's take another example - **a full house** (three of a kind and a pair). This time there are **considerably more possibilities**. After all, any of the `13`

cards in a suit can be the three of a kind, and the pair is in one of the other `12`

cards (it cannot be the same value as the triple). Moreover, the three of a kind are in only three of the four card symbols, and similarly, the pair is in only two.

And that's **where we recall the meaning of a combination**! We need to choose **three out of four** symbols for the triple, and a combination of **two out of four** for the pair. The `n`

choose `k`

formula translates this into `4`

choose `3`

and `4`

choose `2`

, and the binomial coefficient calculator counts them to be `4`

and `6`

, respectively. All in all, if we now **multiply the numbers we've obtained**, we'll find that there are

`13 * 12 * 4 * 6 = 3,744`

possible hands that give a full house. Well, **not too many compared to all the possibilities**, but at least it's `3,744`

times more probable than the royal flash on clubs.

Still, we suggest regularly saving money as a better investment technique than gambling.

## FAQ

### What is the a choose b formula?

The **a choose b formula** is the same as the binomial coefficient formula - it is the factorial of `a`

divided by the product of the factorial of `b`

and the factorial of `a`

minus `b`

. It is also known as the n choose k formula, and can also be solved using Pascal's triangle.

### How do I find 4 choose 2?

To find 4 choose 2:

- Find the factorial of 4 minus 2, which is 2.
- Multiply this number by the factorial of 2, which is also 2, giving 4.
- Divide the factorial of 4, 24, by the number from the previous step, 4.
- The
**result of 4 choose 2 is 6**.

### How do I find 6 choose 2?

To find 6 choose 2:

- Calculate the factorial of 6 minus 2, which is 24.
- Multiply 24 by 2 factorial, which gives 48.
- Work out 6 factorial, which is 720.
- Divide 720 by 48, producing
**15**.

### How are binomial coefficient and Pascal's triangle related?

The **binomial coefficient and Pascal's triangle are intimately related**, as you can find every binomial coefficient solution in Pascal's triangle, and can construct Pascal's triangle from the binomial coefficient formula. For n choose k, visit the n plus 1-th row of the triangle and find the number at the k-th position for your solution.