Count outcomes with the fundamental counting principle, and distinguish permutations (order matters) from combinations (order does not matter) (Statistics and Probability).

What this topic is asking

Counting problems ask how many ways something can happen. The ACT tests the fundamental counting principle (multiply the choices at each stage) and the distinction between permutations (order matters) and combinations (order does not matter). Deciding which applies is the central skill.

The fundamental counting principle

Most ACT counting questions are solved by multiplying.

So with 3 shirts, 2 pants and 4 hats, there are $3 \times 2 \times 4 = 24$ outfits. Multiply, do not add, when each stage's choice combines with the others.

Permutations: order matters

A permutation counts ordered arrangements.

Permutations apply to rankings, finishing orders, distinct roles (president then vice-president), and seatings, anywhere swapping two items gives a different outcome.

Combinations: order does not matter

A combination counts selections where order is irrelevant. Choosing 2 people from 6 to form a committee (no roles) is a combination: the pair {Ann, Bob} is the same as {Bob, Ann}. There are $\frac{6 \times 5}{2} = 15$ such pairs, where the $\div 2$ removes the double-counting of the two orders of each pair. In general a combination divides the permutation count by the number of ways to arrange the chosen items, so it is always less than or equal to the matching permutation count.

Telling them apart

The decision rule is simple but essential: ask whether rearranging the same chosen items produces a different result.

• If yes (order matters), it is a permutation: rankings, sequences, distinct positions.
• If no (order does not matter), it is a combination: groups, committees, hands of cards, selections.

"Choose a president and a treasurer" is a permutation (the roles differ); "choose two officers" with no specified roles is a combination. Reading for words like "arrange", "order", "rank" (permutation) versus "group", "select", "committee" (combination) usually settles it.

Why the order question is everything

Nearly every counting error comes from choosing the wrong model. The reliable test is the rearrangement question above. For straightforward "how many ways" problems with stages, the counting principle (multiply) is enough. When selecting a subset, decide order-matters (permutation) or order-irrelevant (combination) before computing. Getting that one decision right, then multiplying carefully, secures these points.

Try this

Q1. A password uses 2 letters followed by 1 digit, with repetition allowed (26 letters, 10 digits). How many are possible? [1 point]

• Cue. $26 \times 26 \times 10 = 6760$ by the counting principle.

Q2. How many ways can 3 finishers (gold, silver, bronze) be chosen from 5 runners? [1 point]

• Cue. Order matters (distinct medals): $5 \times 4 \times 3 = 60$.