A permutation counts arrangements where order matters, and a combination counts selections where order does not. Ask one question before any calculation: if I swap two of the chosen items, is it a different outcome? This decision appears at the start of every question in permutations and combinations.
How do the two formulas differ?
For n different items, choosing r of them:
- nPr = n × (n − 1) × … with r factors. For example, 8P3 = 8 × 7 × 6 = 336.
- nCr = nPr / r!. For example, 8C3 = 336 / 6 = 56.
The division by r! removes the different orders of the same group. Three people can be put in order in 3 × 2 × 1 = 6 ways, so every group has been counted 6 times in the permutation.
A reliable method
- Identify what is being chosen (people, digits, letters, cards).
- Swap two items in your head. A new outcome means order matters, so use nPr. The same outcome means order does not matter, so use nCr.
- Check repeats. The formulas above assume each item is used at most once. If the question allows repeats, use the multiplication principle instead.
- Calculate by cancelling, then sense-check that nCr is smaller than nPr.
Worked example
A club has 8 members. Find the number of ways to (a) choose a president, a secretary and a treasurer, and (b) choose a committee of 3.
Part (a): the three roles are different, so swapping Aini and Ben changes who is president. Order matters.
8P3 = 8 × 7 × 6 = 336
Part (b): a committee of 3 is the same committee whichever order the names are written. Order does not matter.
8C3 = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56
Check: each committee can fill the three roles in 3! = 6 ways, and 56 × 6 = 336. The answers agree.
The mistake to watch for
A common slip is to use the permutation because it feels more “complete”.
Question: In how many ways can 4 books be chosen from 9 different books to take on a trip?
Mistaken answer: 9P4 = 9 × 8 × 7 × 6 = 3024
The books are only being chosen, not placed in a sequence, so taking Algebra, Biology, Chemistry, Physics is the same as taking Physics, Chemistry, Biology, Algebra. The 3024 counts each group of 4 books 4! = 24 times.
Correction: 9C4 = 3024 / 24 = 126.
Check yourself
Decide whether order matters first, then calculate.
1. In how many ways can 2 students be chosen from 6 to be monitors with the same duties?
Show answer
Same duties, so order does not matter. 6C2 = (6 × 5) / 2 = 15.
2. Six runners are in a race. In how many ways can gold, silver and bronze be awarded?
Show answer
The three medals are different, so order matters. 6P3 = 6 × 5 × 4 = 120.
3. A coach chooses a team of 5 from 11 players. How many different teams are possible?
Show answer
Only the set of players matters. 11C5 = (11 × 10 × 9 × 8 × 7) / 120 = 55 440 / 120 = 462.
Where this leads next
Once the decision is automatic, move on to counting arrangements with a fixed position, then test yourself on the mixed practice set. The non-calculator working trainer helps with the cancelling.
Some students can do every calculation here but misread which situation a long question describes. A teacher on online one-to-one Additional Mathematics tuition can look at exactly how you read the wording.