Q.Find the number of different signals that can be generated by arranging at least 4 flags in order (one below the other) on a vertical staff, if 5 different flags are available.

What is a Factorial?

Imagine you have 3 different books and you want to arrange them on a shelf. How many different ways can you do it?

You could try listing them: Book A, B, C — or A, C, B — or B, A, C — and so on. If you actually do this, you'll find exactly 6 arrangements.

That "6" is no accident. It comes from multiplying: $3 \times 2 \times 1 = 6$.

This pattern — multiplying a number by every positive integer smaller than it, all the way down to 1 — is called a factorial. We write it with an exclamation mark: $3! = 3 \times 2 \times 1 = 6$.

The Precise Definition

For any positive integer $n$, the factorial $n!$ is defined as:

So:

• $1! = 1$
• $2! = 2 \times 1 = 2$
• $3! = 3 \times 2 \times 1 = 6$
• $4! = 4 \times 3 \times 2 \times 1 = 24$
• $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

Notice how fast factorials grow. $5!$ is already 120, and $10!$ is 3,628,800.

The Special Case: $0!$

This often confuses students. What is $0!$ ?

By the definition above, $0!$ would be an empty product — multiplying nothing. That doesn't make sense directly, so we define it by what makes the mathematics consistent.

Consider this pattern:

$\frac{1!}{1} = 0! \quad \Rightarrow \quad \frac{1}{1} = 1$ …