Q.There are 5 true-false questions in a test. If no two students have answered the same sequence of answers and no student has given all correct answers. How many students are there is the class for this to happen?

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:

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