Falling and rising factorials facts for kids

The falling factorial and rising factorial are special ways to multiply numbers. They are like a series of multiplications, but the numbers either go down or up in a specific pattern.

What Are Factorials?

Factorials are a fun way to multiply numbers. Usually, when you hear "factorial," it means multiplying a number by all the whole numbers smaller than it, down to 1. For example, 5 factorial (written as 5!) is 5 × 4 × 3 × 2 × 1 = 120.

But falling and rising factorials are a bit different. They don't always go all the way down to 1 or start from 1. Instead, they involve a specific number of steps.

Falling Factorials

A falling factorial starts with a number and then multiplies it by numbers that are one less, each time. You do this for a certain number of steps.

Imagine you have a number, let's call it n. You want to find its falling factorial for k steps. This means you multiply n by (n - 1), then by (n - 2), and so on, until you have multiplied k times.

How Falling Factorials Work

Let's look at an example. If n is 7 and k is 3, the falling factorial would be:

• Start with 7.
• First step: 7 × (7 - 1) = 7 × 6
• Second step: (7 × 6) × (7 - 2) = 7 × 6 × 5
• Third step: (7 × 6 × 5) × (7 - 3) = 7 × 6 × 5 × 4

Wait, that's not right! The definition says "k number of terms". Let's re-explain: If n is 7 and k is 3, you take 3 terms starting from 7 and going down by 1 each time.

• The first term is 7.
• The second term is 7 - 1 = 6.
• The third term is 7 - 2 = 5.

So, the falling factorial of 7 for 3 terms is 7 × 6 × 5 = 210.

It's like counting down for a few steps and multiplying those numbers together.

Rising Factorials

A rising factorial is the opposite of a falling factorial. It starts with a number and then multiplies it by numbers that are one more, each time. You also do this for a specific number of steps.

If you have a number n and you want its rising factorial for k steps, you multiply n by (n + 1), then by (n + 2), and so on, until you have multiplied k times.

How Rising Factorials Work

Let's use an example for rising factorials. If n is 4 and k is 3, the rising factorial would be:

• The first term is 4.
• The second term is 4 + 1 = 5.
• The third term is 4 + 2 = 6.

So, the rising factorial of 4 for 3 terms is 4 × 5 × 6 = 120.

It's like counting up for a few steps and multiplying those numbers together.

Why Are They Used?

Falling and rising factorials are used in different areas of math, especially in topics like:

• Combinatorics: This is the study of counting and arrangements. These factorials help count how many ways things can be arranged or chosen.
• Calculus: They appear in advanced math when dealing with differences between numbers in a sequence.
• Probability: Sometimes, they help calculate the chances of certain events happening.

They are special tools that make certain calculations easier in higher-level mathematics.

See also

In Spanish: Factoriales descendente y ascendente para niños