AI Tutor
If a smartphone requires a standard 4-digit passcode, how many unique codes could someone possibly enter before cracking it?
Instead of writing down thousands of combinations by hand, we can count the total outcomes systematically using a core rule of combinatorics.

The Fundamental Counting Principle

An outcome is a single possible result of an experiment or process, while an event is any specific collection of one or more outcomes.
The Fundamental Counting Principle states that if a first choice can be made in n1n_1 ways, and a second choice can be made in n2n_2 ways, the total number of sequential outcomes is n1×n2n_1 \times n_2.
In 1654, French mathematicians Pierre de Fermat and Blaise Pascal analyzed games of chance and laid down this systematic multiplication rule to count outcomes without exhaustive listing.
Why does multiplication work here instead of addition? Every single choice in the first stage branches into an identical set of options in the next stage.

Why the Multiplication Rule Works

Consider n1n_1 groups, where each group independently contains n2n_2 distinct items. Grouping creates n1n_1 sets of size n2n_2, which yields n1×n2n_1 \times n_2 total elements by the definition of multiplication.
For kk sequential independent steps, mathematical induction extends this logic: the total number of possible outcomes is the product of the choices at every step, N=n1×n2×n3×⋯×nkN = n_1 \times n_2 \times n_3 \times \dots \times n_k.
Now let's apply this principle to multi-stage real-world scenarios. How many 4-digit passcodes can we form when digits are allowed to repeat?