Counting combinations — Mathematics, 14–17
How to count possible selections without writing every choice down. Mathematics, 14–17 years.
Selections, not arrangements
A combination is a choice where order does not matter. Choosing Ana, Ben and Chi is the same group as choosing Chi, Ana and Ben. The notation C(n,r) counts how many groups of r objects can be selected from n objects.
Why count systematically?
When choices multiply, listing them invites omissions and duplicates. A football coach selecting 3 players from 10 needs the number of different teams, not the order in which names were written. Combinations grew from this practical problem and also provide the counts behind probability calculations.
Choosing a committee
How many 3-person committees can come from 5 people? Start with ordered choices: 5 × 4 × 3 = 60. Each committee appears 3 × 2 × 1 = 6 times, once for each ordering. Divide: 60 ÷ 6 = 10. Equivalently, C(5,3) = 5!/(3!2!) = 10.
Do not count the order twice
A common mistake is to count ABC, ACB and BAC as different committees. That is reasonable because they are different writing orders, but the committee itself has not changed. First decide what the question calls different; remove repeated orderings only when order truly does not matter.
Planning real choices
Combinations help plan teams, panels, research samples and sets of lottery tickets. A scientist choosing 4 test sites from 20 can know how many samples are possible before collecting data. The method is honest only when every object can be chosen and the order of selection has no meaning.
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