Q.Determine the number of paths in the xy-plane from (1, 2) to (7, 5), where each such path is made up of individual steps going one unit to the right (R) or one unit upwards (U). [Figure: a coordinate grid with a marked point at (1,2) and another at (7,5); a path consists of a sequence of unit rightward and unit upward steps joining the two points.]

The Fundamental Principle of Counting

Imagine you're getting dressed. You have 3 shirts (red, blue, green) and 2 pairs of pants (black, white). How many different outfits can you make?

You could list them: red-black, red-white, blue-black, blue-white, green-black, green-white. That's 6 outfits.

Notice something: 3 shirts × 2 pants = 6 outfits. That's not a coincidence — it's the entire idea.

The Intuition: A Fork in the Road

Think of each choice as a fork in a path. First you pick a shirt: 3 options. For each shirt, you then pick pants: 2 options. So the total number of paths through the whole process is 3 × 2 = 6.

This works no matter how many stages you add. If you also had 2 pairs of shoes, the total outfits become 3 × 2 × 2 = 12. Each new choice multiplies the total.

The Precise Statement

Fundamental Principle of Counting (Multiplication Principle)

If an event can happen in $m$ ways, and after it happens a second event can happen in $n$ ways, then the two events together can happen in $m \times n$ ways.

More generally: if a task consists of $k$ steps, and step 1 can be done in $n_1$ ways, step 2 in $n_2$ ways, ..., step $k$ in $n_k$ ways, then the total number of ways to complete the entire task is:

A Simple Example

How many 2-digit numbers can you form using the digits 1, 2, 3, 4, 5 if repetition is allowed? …