Binomial coefficient

What is a binomial coefficient?

The binomial coefficient, written as $\binom{n}{k}$ and pronounced "n choose k," represents the number of ways to select k items from a group of n items, where the order doesn't matter.

It's calculated using the formula:

\binom{n}{k} = \frac{n!}{k!(n-k)!}

Where $n!$ (read as "n factorial") means multiplying all whole numbers from n down to 1.

Simple examples

Example 1: Selecting a committee

If you need to choose 3 people from a group of 8 people to form a committee, how many different possible committees could you create?

\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} = \frac{8 \times 7 \times 6 \times 5!}{3 \times 2 \times 1 \times 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56

So there are 56 different possible committees.

Example 2: Card hands

How many different 5-card hands can be dealt from a standard 52-card deck?

\binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!} = 2,598,960

That's why poker has so many possibilities!

Practical applications

1. Probability

Binomial coefficients are used to calculate probabilities in situations where there are "success" and "failure" outcomes.

For example, if you flip a fair coin 10 times, the probability of getting exactly 6 heads is:

P(6 \text{ heads}) = \binom{10}{6} \times \left(\frac{1}{2}\right)^6 \times \left(\frac{1}{2}\right)^4 = 210 \times \frac{1}{1024} \approx 0.205

2. Expanding binomials

When you expand expressions like $(x + y)^n$ , the binomial coefficients appear as the coefficients:

(x + y)^4 = \binom{4}{0}x^4y^0 + \binom{4}{1}x^3y^1 + \binom{4}{2}x^2y^2 + \binom{4}{3}x^1y^3 + \binom{4}{4}x^0y^4

Which simplifies to:

(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4

3. Computer science

Binomial coefficients help in analyzing algorithms, particularly in understanding how many different ways operations can be performed.

4. Sports and gaming

They're used to calculate possible outcomes in tournaments, bracket systems, and game theory.

Pascal's triangle

You can find binomial coefficients in Pascal's triangle:

Each number is $\binom{n}{k}$ where n is the row number and k is the position (both starting from 0).

For example, in row 4, the numbers 1, 4, 6, 4, 1 are $\binom{4}{0}$ , $\binom{4}{1}$ , $\binom{4}{2}$ , $\binom{4}{3}$ , and $\binom{4}{4}$ .

Useful properties

Symmetry: $\binom{n}{k} = \binom{n}{n-k}$

This means choosing k items is the same as choosing which (n-k) items to leave out.
Edge cases: $\binom{n}{0} = \binom{n}{n} = 1$

There's only one way to choose 0 items (choose nothing), and only one way to choose all n items.
Building Pascal's triangle: $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$

Each number is the sum of the two numbers directly above it in Pascal's triangle.

Why binomial coefficients matter

Binomial coefficients are important because they help us count possibilities and calculate probabilities in many real-world situations. From designing experiments to analyzing data, understanding these coefficients gives you a powerful tool for solving problems involving combinations and selections.

Whether you're studying statistics, computer science, or just trying to figure out lottery odds, binomial coefficients provide a fundamental way to understand how many different ways things can be arranged or selected.

Binomial Coefficient Calculator

Formula

References