Pascal's Triangle

Pascal’s triangle is a number pattern that fits in a triangle. It is named after Blaise Pascal, a French mathematician, and it has many beneficial mathematic and statistical properties, including finding the number of combinations and expanding binomials.

To make Pascal’s triangle, start with a 1 at that top. Then work your way down in a triangular pattern. Each value in the triangle is the sum of the two values above it. The animation below depicts how to calculate the values in Pascal’s triangle.

Navigating Pascal’s Triangle

The notation for Pascal’s triangle is the following:

n = row the number. The top of the pyramid is row zero. The next row down with the two 1s is row 1, and so on.
k = the column or item number. K = 0 for the left-most values and increases by one as you move right.

The notation for an entry in Pascal’s triangle at row n and column k is the following:

${n \choose k}$

For example:

${6 \choose 3}= 20$

Caution, it’s easy to forget that the top of the triangle is row 0 and that the first 1 in any row is item or column 0.

Using Pascal’s Triangle to Find the Number of Combinations

This number pattern has many intriguing and valuable properties. Because this is a statistics blog, I’ll start with its ability to find combinations. In probability theory, combinations are a sequence of outcomes where order does not matter. For example, a pizza with ham, mushroom, and pepperoni is a combination. You can change the order of those ingredients, but it’s still the same pizza.

When calculating probabilities, you’ll often need to find the number of combinations given several parameters. The standard notation for combinations is the following

nCr

Where:

n = the number of options
r = the size of the combination

You can use Pascal’s triangle to find the number of combinations without repetition, which means the outcomes cannot repeat. To use Pascal’s triangle to find the number of combinations, look in row n, column r.

Suppose we want to find the number of pizza combinations using five possible ingredients (n = 5), and we’ll only include three on the pizza (r = 3). And you can only use each ingredient once—no double pepperoni!

To use Pascal’s triangle to find the number of combinations for ₅C₃, look in row 5, column 3.

${5 \choose 3}= 10$

There are 10 combinations for the specified parameters!

Pascal’s Triangle and Binomial Expansion

In algebra, binomial expansion describes expanding (x + y)ⁿ to a sum of terms using the form ax^by^c, where:

b and c are nonnegative integers
n = b + c
a = is the coefficient of each term and is a positive integer.

For example, (x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

Notice that the coefficients in the equation are: 1, 4, 6, 4, 1.

Using Pascal’s triangle, you can find the coefficient values of a binomial expansion by looking at row n, column b. For our example, n = 4 and b ranges from 4 to 0.

For our example binomial expansion, we need to look at the 4^th row. Then work our way through the b values, 4 to 0. Voila! Pascal’s triangle provides the coefficients for the binomial expansion!

Other Uses for Pascal’s Triangle

Because my site is primarily about statistics and probability, I’ll only touch on several other patterns and ways to use Pascal’s triangle.

Prime numbers

When the 1^st element of a row (the first number after the leading 1) is a prime number, you can evenly divide all numbers in that row (except the 1s) by it.

Natural numbers, triangular numbers, and more!

When you left justify Pascal’s triangle, the columns represent various types of numbers.

By Cmglee – Own work, CC BY-SA 4.0

Fibonacci sequence

When you left justify the rows, the diagonals in Pascal’s triangle sum to the Fibonacci sequence. Learn more about the many interesting uses for the Fibonacci sequence!

By Phan Yamada – Own work, CC BY-SA 4.0

Powers of 2

The sum of each row equals 2ⁿ, where n = the row number.

Hockey Stick Pattern

Start at any of the 1s at either edge of the triangle. Work your way down a diagonal. At any point, bend your path downward. That last value equals the sum of the previous values.