Q.If $\dfrac{1}{6!} + \dfrac{1}{7!} = \dfrac{x}{8!}$, find $x$.

Factorial Arithmetic — From Intuition to Precision

Imagine you have 3 different books you want to arrange on a shelf. How many different ways can you line them up? You could try listing them: Book A, B, C — or A, C, B — or B, A, C — and so on. If you actually count, you'll find 6 arrangements.

Where does that 6 come from? For the first position, you have 3 choices. Once you pick one, you have 2 choices left for the second position. Then only 1 choice remains for the last spot. So the total is $3 \times 2 \times 1 = 6$.

That product — multiplying a whole number by every positive integer smaller than it, all the way down to 1 — is called a factorial. It's one of the most useful shortcuts in counting.

The symbol is an exclamation mark: $n!$ is read as "n factorial". It only makes sense for non-negative integers.

The first few values

The recursive nature

Factorials have a beautiful pattern: every factorial is the current number times the previous factorial.

This recursive definition is often how you'll compute factorials in problems: $n! = n \times (n-1)!$, with the base case $0! = 1$.

Why factorials explode so fast

Notice how quickly the numbers grow: $5! = 120$, $6! = 720$, $7! = 5040$, $10! = 3,628,800$. By $20!$, you're already at 2.4 quintillion. This rapid growth is why factorials appear in probability (counting arrangements of decks of cards), combinatorics (choosing teams), and even in advanced mathematics like Taylor series.

The core idea in one sentence …