Binomial Coefficient: Formula & Examples

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  • 0:00 The Binomial Coefficients
  • 1:05 Formula for Binomial…
  • 3:07 Pascal's Triangle
  • 4:09 A Card Counting Example
  • 5:09 Lesson Summary
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Lesson Transcript
Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

In this lesson, you will discover the binomial coefficients, learn how to compute them, and find out what they can be used for. Some examples will highlight how they are used in counting problems.

The Binomial Coefficients

The easiest way to explain what binomial coefficients are is to say that they count certain ways of grouping items. Specifically, the binomial coefficient C(n, k) counts the number of ways to form an unordered collection of k items chosen from a collection of n distinct items.

For example, if you wanted to make a 2-person committee from a group of four people, the number of ways to do this is C(4, 2). Incidentally, there are 6 ways - just label the people A, B, C, D, and list all the two-letter sets: AB, AC, AD, BC, BD, and CD (remember, order doesn't matter, so BA is not any different than AB). So, we now know that C(4, 2) = 6.

There are many different notations used for the binomial coefficients. All of these are equivalent to C(n, k) and may be read as: n choose k. The last notation (the one that looks a bit like a fraction) is the one most commonly used, though you will see C(n, k) fairly often, because it is easier to type.

Different notations for the binomial coefficient

Formula for Binomial Coefficients

Okay, so how do we compute binomial coefficients? As you might have guessed, there is a formula:

Binomial coeffients formula

The exclamation points are actually part of the formula (and they don't mean the numbers are excited). The notation n! is called the factorial of n, and it means to multiply n times (n - 1) times (n - 2), times every whole number down to 1. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120. Just be careful of one special case: 0! = 1, by definition.

On the other hand, most people will end up using the second form of the formula, in which the multiplications are written out more explicitly (and some cancellation has already been done for you). Let's practice using both versions of the formula.

Example 1

Compute C(7,3) using two different ways.

We have n = 7 and k = 3, and we are computing 7 choose 3.

Using factorials:

7 choose 3

To use the shortcut formula, first find out the value of (n - k + 1), which is (7 - 3 + 1), which is 5. This will be the last factor on the top (or in the numerator) of the fraction. By the way, there will always be the same number of factors on the top (numerator) as the bottom (denominator).

7 choose 3

By the way, this means there are exactly 35 ways to form a 3-person committee from a pool of 7 people.

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