Counting problems appear on nearly every ACT. The SAT skips them entirely, making this a must-know ACT topic. The good news: there is one simple question that tells you which formula to use.
The One Question That Decides Everything
Ask yourself: "Does the order of selection matter?"
- Yes → Permutation (arrangements, rankings, positions)
- No → Combination (groups, teams, selections)
The Formulas
Permutation (order matters):
Combination (order does not matter):
The only difference is the in the denominator. Combinations divide out the duplicate arrangements.
Example 1: Permutation
From 10 runners, how many ways can gold, silver, and bronze be awarded?
Order matters (gold is different from silver), so:
Example 2: Combination
From 10 people, how many ways can a committee of 3 be chosen?
Order does not matter (a committee of Alice, Bob, Carol is the same as Carol, Alice, Bob):
Example 3: With Repetition
How many 4-digit PIN codes are possible using digits 0-9?
Each position has 10 choices independently:
This is neither a permutation nor combination — it is the multiplication principle with repetition allowed.
Common Mistakes to Avoid
- Using permutations when order does not matter: Choosing a committee is a combination, not a permutation. If you get an answer that seems too large, you probably used the wrong formula.
- Forgetting to divide by : The difference between and is exactly . If your answer is 6 times too large, you forgot to divide by .
- Confusing "with repetition" and "without": Passwords allow repetition. Choosing team members from a class does not (one person cannot be chosen twice).
ACT Pro Tip
On the ACT, the most common counting problems involve choosing committees (combination), arranging people in seats (permutation), or counting passwords/codes (multiplication principle). Identify the type in the first 5 seconds, then apply the right formula.
Master counting with our Permutations and Combinations lesson — ACT-exclusive content with 8 practice exercises.