Year 12 Probability: Combinations

Combinations are used when a probability question is about selecting a group and the order does not matter. They are common in committee, card and lottery-style problems.

If choosing A then B gives the same group as choosing B then A, use combinations rather than permutations.

Core idea

The number of ways to choose $r$ objects from $n$ distinct objects is $C (n, r) = n! / (r! (n - r)!)$ .

Worked example

Problem: A committee of 3 students is chosen from 6 girls and 4 boys. What is the probability that the committee contains exactly 2 girls?

Total committees = $C (10, 3) = 120$ .
To get exactly 2 girls, choose 2 of the 6 girls and 1 of the 4 boys.
Favourable committees = $C (6, 2) \times C (4, 1) = 15 \times 4 = 60$ .
Probability = $60 / 120 = 1 / 2$ .

Answer: the probability of choosing exactly 2 girls is

1 / 2

When it is used

Selecting committees or teams.
Choosing cards from a deck.
Lottery and sample selection problems.
Probability questions where grouping matters but order does not.

Worked example 2

Problem: Two cards are chosen from a standard deck of 52 cards without replacement. What is the probability that both cards are aces?

Total 2-card selections: $C (52, 2)$ .
Favourable selections: choose 2 of the 4 aces, so $C (4, 2)$ .
Therefore $P (both aces) = C (4, 2) / C (52, 2)$ .
Compute: $6 / 1326 = 1 / 221$ .

Answer: the probability is

1 / 221

Common mistakes

Using permutations for committee-style questions.
Counting the same group several times in different orders.
Forgetting that the sample space and the favourable cases must be counted in the same way.

Revision focus

Probability work becomes far more reliable when you define the object of the question before calculating. Is the problem about ordered outcomes, unordered selections, a conditional event, a discrete random variable, or a continuous density? That decision controls the method. In many exam questions, the major error is not arithmetic. It is using a correct formula for the wrong kind of probability situation.

A second useful habit is to interpret the result immediately after it is found. Decide whether the answer is a count, a probability, an average outcome, or a spread measure. In continuous settings, remember that probabilities come from areas over intervals, not from single points. In discrete settings, remember that the total probability must still add to one. Those interpretation checks are often the fastest way to catch an error before it reaches the final line.

In probability, the final line should always answer the right event, not just display a calculation. After computing, read the question again and confirm that your sample space, condition, or random-variable interpretation still matches what was asked. That last comparison catches a surprising number of near-miss errors.

When the same collection can be chosen in several orders but still counts as one group, combinations are almost always the cleaner and more economical language for the problem.