Q.Evaluate $\dfrac{8!}{6!\,2!}$.

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:

So $0! = 1$. This isn't a guess — it's forced by the pattern, and it makes all formulas involving factorials work correctly.

Why Do We Need Factorials?

Factorials are the building block for counting arrangements and selections. The two most common uses are:

Permutations — arranging $n$ distinct objects in order: $n!$ ways.

Combinations — choosing $r$ objects from $n$ without caring about order: $\frac{n!}{r!(n-r)!}$.

You'll see these in probability, statistics, and algebra (binomial theorem). For now, just get comfortable with computing factorials and understanding that $n!$ counts the number of ways to order $n$ distinct items.

A Quick Check

Try these yourself:

1. $6! = ?$
2. $\frac{7!}{5!} = ?$
3. $4! + 3! = ?$

Answers: 720, 42, 30. If you got them, you've understood the core idea.