Combinations & Permutations: Counting When Order Matters (or Doesn't)

Counting problems sound deceptively simple until you have to decide: is choosing Anna, then Ben the same as choosing Ben, then Anna? For a two-person committee, yes — same committee. For first and second place in a race, no — different outcomes. That single distinction, whether order matters, is the entire difference between a combination and a permutation, and once you can spot it, the formulas are mechanical.

This guide builds from factorials to both formulas and shows how to tell them apart, with worked and practice problems matched to real test difficulty at Northside Tutoring.

Why Counting Matters

Combinations and permutations extend the basic counting principle and feed directly into probability. On the SAT and ACT they appear as committee, arrangement, and selection problems. They're Algebra II content beyond the SSAT.

The One Question: Does Order Matter?

Before any formula, ask: if I rearrange the same chosen items, is that a different outcome? If yes (a ranking, a sequence, a seating order), it's a permutation. If no (a team, a group, a handful), it's a combination. Everything follows from this.

Factorials: The Building Block

A factorial, written n!, is the product of all whole numbers from n down to 1: 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials count the number of ways to arrange n items in order, and they're the engine inside both formulas. (By definition, 0! = 1.)

Permutations (Order Matters)

The number of ways to arrange r items chosen from n, when order matters:

To award gold, silver, and bronze among 5 runners: 5P3 = 5!/2! = 120/2 = 60 ways.

Combinations (Order Doesn't)

The number of ways to select r items from n, when order doesn't matter:

To pick a 3-person committee from 5 people: 5C3 = 5!/(3!2!) = 120/(6·2) = 10 ways. Notice it's smaller than the permutation, because the formula divides out the r! ways of ordering the same group.

Telling Them Apart

Permutation (order matters)	Combination (order doesn't)
rankings, finishing order	committees, teams
passwords, license plates	choosing a subset
seating arrangements	hands of cards

Where You'll See This — Test by Test

No reference sheet provides these formulas. The SAT and ACT test choosing between combinations and permutations and applying the right one. It's Algebra II material beyond the SSAT.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

Every Northside student works through a four-step framework:

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