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Math · Statistics & Probability

Counting problems (combinations and permutations)

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.

Practise counting problems (combinations and permutations) nowSee sample questions
3
Difficulty tiers
40s
Target pace
47%
Average first-attempt score

What's covered

  • Permutations — count arrangements where order matters, using nPr = n!/(n−r)! for selecting r items from n.
  • Combinations — count selections where order doesn't matter, using nCr = n!/(r!(n−r)!) — always smaller than the corresponding permutation count.
  • The fundamental counting principle — multiply the number of choices at each independent step to find the total number of outcomes.
  • Permutations with repetition or restrictions — adjust the basic formula when some items are identical, or when specific items must occupy specific positions.
  • Recognizing which one a word problem needs — look for language like "arrange," "order," or "first/second/third place" for permutations, versus "choose," "select," or "committee" for combinations.

Where students lose marks

Using nPr when the problem doesn't care about order

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.

Using nCr when the problem does care about order

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.

Forgetting to divide out repeated items

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.

Three sample questions

Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.

Sample 01Foundation

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?

Sample 02Core

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?

Sample 03Advanced

How many distinct arrangements are there of the letters in the word LEVEL?

How to practise this

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.

State whether order matters before picking a formula

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.

Then 40 seconds a question

Move to timed sessions once the order-matters decision is instant. Problems with repeated items or restrictions run a bit longer.

Fold into a mock

Pair counting problems with basic probability in a mock — many probability sample spaces are counted using exactly these formulas.

Stop guessing the formula.
Start deciding if order matters.

  • An order-matters walkthrough on every item
  • Timed or untimed sessions, any length you like
  • Tracked separately — see your counting problems accuracy and pace over time
Start practising free

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Basic probabilityMean, Median, ModeAll of Math