Decide whether order matters, then count arrangements or selections without listing them all out — that one decision determines which formula the rest of the problem depends on. Every question comes with a written explanation of why order did or didn't matter.
Selecting a committee, a group, or a hand of cards doesn't care who was picked first — that calls for nCr, and using nPr overcounts every group multiple times.
Assigning distinct roles — president, secretary, treasurer — to a group of people cares about which person gets which role, so that calls for nPr, not nCr.
Arranging the letters of a word with repeated letters, like MISSISSIPPI, requires dividing by the factorial of each repeated letter's count — treating every letter as distinct overcounts identical arrangements.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
A club with 8 members needs to choose a president, a vice president, and a treasurer, with no one holding more than one role. In how many ways can this be done?
A — 336. Assigning three distinct roles cares about order — who gets which role — so this is a permutation: 8P3 = 8×7×6 = 336. 56 (B) is 8C3, the combination count — the result of treating the three roles as interchangeable instead of distinct.
A pizza shop offers 6 toppings. How many different 4-topping pizzas can be made, if each topping can only be used once per pizza?
A — 15. A set of toppings doesn't care about the order they're added, so this is a combination: 6C4 = 6!/(4!2!) = 15. 360 (B) is 6P4, the permutation count — the result of treating different topping orders as different pizzas, when a pizza with pepperoni added before mushroom is the same pizza as the reverse.
How many distinct arrangements are there of the letters in the word LEVEL?
A — 30. LEVEL has 5 letters, with L repeated twice and E repeated twice. The total arrangements of 5 distinct letters would be 5! = 120, but that overcounts arrangements that look identical due to the repeats — divide by 2! for the repeated L's and 2! for the repeated E's: 120/(2!×2!) = 120/4 = 30. 60 (C) comes from dividing by only one of the two repeated letters instead of both.
The entire skill in this subtopic is one decision — does order matter — made correctly before any formula gets touched. Practice is about training that judgment call, not memorizing the formulas.
For every problem, decide and say out loud whether rearranging the same items would count as a different outcome. Untimed practice is where that judgment call gets reliable.
Move to timed sessions once the order-matters decision is instant. Problems with repeated items or restrictions run a bit longer.
Pair counting problems with basic probability in a mock — many probability sample spaces are counted using exactly these formulas.
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