Combinations and permutations: choosing the right method

Choosing three books for a reading group sounds simple, but the counting method changes when order becomes part of the task. A set of three books is a combination because rearranging the same titles creates no new outcome. The first three runners crossing the line form a permutation because first place matters, as do second and third.

The same distinction appears in school projects, sports, travel planning and household decisions. The important question is not whether a real-life choice feels “orderly”; it is whether swapping two chosen elements changes the result being counted. Once that test is clear, factorial notation gives a reliable way to calculate both ideas and avoid listing every possibility by hand.

The core difference

A combination counts selections in which order is ignored. The formula ( {}^{n}C_r=\frac{n!}{r!(n-r)!}) chooses (r) items from (n) without arranging them. If eight people apply for a community grant committee and six are chosen, changing the order in which their names are written does not produce another committee. There are ( {}^{8}C_6=28) possible committees.

A permutation counts ordered arrangements. Its formula is ( {}^{n}P_r=\frac{n!}{(n-r)!}). If those eight people fill three distinct positions, such as president, secretary and treasurer, swapping two applicants changes the outcome. Here the count is ( {}^{8}P_3=336). The (r!) in the combination denominator removes the duplicate counting caused by rearranging each chosen group.

Choosing the right method

Start with a swap test: exchange two elements and ask whether the answer has genuinely changed. Choosing three syrups for a Melbourne laneway café gives a combination, while naming the first, second and third place in a café popularity vote gives a permutation. In the first case the flavours form one group; in the second, each position has a different meaning.

Order can also arise through a rule rather than an obvious ranking. The order in which three songs play at a school assembly is irrelevant unless a running order is specified. Conversely, the order of teams in a knockout draw is relevant because the first team drawn receives a different fixture from the second. Read the conditions carefully instead of relying on the wording “choose” or “arrange”.

A selection example

Suppose a Brisbane film society wants three films from eight shortlisted titles for a programme. The viewing panel does not care which film is listed first, so the task is a combination. There are ( {}^{8}C_3=\frac{8!}{3!5!}=56) possible three-film programmes.

If the panel also assigns each film to one of three labelled time slots, order matters. The number becomes ( {}^{8}P_3=\frac{8!}{5!}=336), which is six times as large because every three-film group can be ordered in (3!=6) ways. This comparison often makes the role of order easier to see than the formula alone.

An arrangement example

Imagine a Sydney community group choosing three office-bearers from seven candidates. The positions are different, so selecting the same three people in a different order creates a different leadership team. The count is ( {}^{7}P_3=7\times6\times5=210): seven choices for president, six remaining choices for secretary and five for treasurer.

When a task has two stages, calculate each stage separately. Choosing four swimmers from a club of eight and then arranging them in the medley relay gives ( {}^{8}C_4\times4!=70\times24=1{,}680). The first factor chooses the team; the second determines its running order. This staged approach is often clearer than searching for one formula that hides the two decisions.

Repeated choices and exponential patterns

Repeated decisions do not always mean permutations. If a quiz night has three questions and each has two possible answers, the product rule gives (2\times2\times2=8) answer patterns. Each complete pattern is an outcome, but there is no separate group of people or objects to arrange into positions.

This multiplication is similar to the repeated-factor thinking in CurryPlum’s lesson on growth and decay models. A useful warning follows from that comparison: multiplying by (r) counts (r) choices at each stage, whereas a factorial counts the shrinking number of options available after something is used. If choices may be repeated and each one is recorded, the general pattern is (n^r), provided the (n) options are treated as distinct.

Common traps and special cases

One frequent error is counting ordered arrangements and forgetting to remove the irrelevant order. For instance, ( {}^{n}P_r) divided by (r!) gives ( {}^{n}C_r). Another error is assuming every real-life selection has unlimited choices. A raffle ticket is selected without replacement, while a café order may contain several identical flat whites.

Identical objects need special care. An Australian pocket containing two $1 coins, one 50-cent coin and one 20-cent coin has four different ways to choose two coins when order is ignored, not six. Coins of the same denomination are indistinguishable. If a shop offers unlimited items from five types and a customer chooses three items with types allowed to repeat, the count is ( {}^{5+3-1}C_3={}^{7}C_3=35).

Restrictions can also change the method. For an Australia Day raffle, the number of four-ticket selections containing at least one winner is ( {}^{40}C_4-{}^{36}C_4). At a circular dinner, rotating the entire seating plan does not create a new arrangement, so the standard count for (n) guests becomes ((n-1)!), provided mirror-image seatings count separately.

A practical decision routine

For any problem, identify the objects, decide whether order changes the outcome, and check whether choices may repeat. Then label the task before calculating. “Choose three badges for display” and “appoint a president, secretary and treasurer” may look similar, but only the second treats position as significant. A clear step-by-step solution should make those choices visible, whether it comes from class notes, a calculator or online homework assistance.

You can rehearse that routine with interactive maths practice, using a fresh set of people, objects or positions each time. For a Canberra market stall, a volunteer might select two products from a Woolworths range and then place them in distinct positions on the table. Calculating the selection first and the arrangement second keeps the assumptions visible and makes errors easier to trace.

Your next step is to solve one combination problem and one permutation problem, label which method each requires, and check both answers against the order test.