AI Tutor
If your smartphone uses a simple 4-digit numeric passcode, there are 10,000 possible codes. How do just four small positions produce such a massive pool of possibilities?
To answer this, mathematicians use the multiplication principle, which calculates the total outcomes when decisions occur across sequential, independent stages.
In 1654, French mathematicians Blaise Pascal and Pierre de Fermat formalized these foundational counting rules while analyzing multi-stage games of chance.

The Fundamental Counting Principle

If a first stage has mm possible outcomes and a second independent stage has nn possible outcomes, the combined process has m×nm \times n total outcomes.
Why does this work? Each initial choice sprouts its own identical branch of nn secondary choices, creating mm groups of size nn.
How does this rule extend when you move from two choices to three, four, or more sequential steps?
The Multiplication Counting Principle (the Fundamental Counting Principle for independent stages) states that across kk successive stages with n1,n2,…,nkn_1, n_2, \dots, n_k outcomes, the total number of combined outcomes is N=n1×n2×⋯×nkN = n_1 \times n_2 \times \dots \times n_k.

Multi-Stage Passcodes

Suppose you create a 3-letter code using standard English letters (A through Z). If letters can repeat, each of the three slots has 26 independent options, giving 26×26×26=17,57626 \times 26 \times 26 = 17,576 total codes.
If repetition is forbidden, your choices shrink at each step: 26 options for the first letter, 25 for the second, and 24 for the third, yielding 26×25×24=15,60026 \times 25 \times 24 = 15,600 total codes.