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AMC 10 & 12

Permutations & Combinations

Does order matter? That single question decides which formula to use — get it right and the rest is arithmetic.

Permutations: order matters

The number of ways to arrange kk items chosen from nn distinct items, in order:

P(n,k)=n!(n−k)!=n(n−1)(n−2)⋯(n−k+1)P(n,k) = \dfrac{n!}{(n-k)!} = n(n-1)(n-2)\cdots(n-k+1)

Combinations: order doesn't matter

The number of ways to choose kk items from nn, ignoring order:

C(n,k)=(nk)=n!k!(n−k)!=P(n,k)k!C(n,k) = \binom{n}{k} = \dfrac{n!}{k!(n-k)!} = \dfrac{P(n,k)}{k!}

The k!k! in the denominator is exactly the overcounting factor — every unordered group ofkk items corresponds to k!k! different ordered arrangements, so dividing removes the duplicate orderings.

Variants worth knowing

Permutations with repetition allowed: nk\text{Permutations with repetition allowed: } n^k
Circular permutations of n items: (n−1)!\text{Circular permutations of } n \text{ items: } (n-1)!
Multinomial (arranging n items with repeats): n!n1! n2!⋯nr!\text{Multinomial (arranging } n \text{ items with repeats): } \dfrac{n!}{n_1!\,n_2!\cdots n_r!}

Circular permutations use (n−1)!(n-1)! instead of n!n! because rotating a circular arrangement doesn't create a new arrangement — fixing one item's position removes that redundancy.

Worked example

Problem: A committee of 3 people is chosen from a group of 8, and then one of the 3 is designated chairperson. In how many ways can this be done?

Solution: First choose the committee (order doesn't matter): (83)=56\binom83=56ways. Then choose the chairperson from those 3: 33 ways. Total:

56×3=16856 \times 3 = 168

Alternatively: choose the chairperson directly from all 8 (88 ways), then the remaining 2 committee members from the other 7 ((72)=21\binom72=21 ways): 8×21=1688 \times 21 = 168 — same answer, different route.

Practice problems

1.In how many ways can 4 runners finish 1st, 2nd, 3rd, and 4th out of a race of 10 runners?

2.How many 5-person committees can be formed from a group of 12?

3.In how many distinct ways can 6 people be seated around a circular table?

4.How many distinct arrangements are there of the letters in MISSISSIPPI?

Counting PrinciplesStars and Bars