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Statistics

Permutations and Combinations

Lesson

Many probability questions need counts of arrangements or selections. Three tools to know:

Factorial — how many ways to arrange nn distinct items in a row:

n!=n⋅(n−1)⋅(n−2)⋯2⋅1n! = n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1

Example: 5!=5⋅4⋅3⋅2⋅1=1205! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120. By convention 0!=10! = 1.

Permutations — number of arrangements of kk items chosen from nn, where order matters:

P(n, k)=n!(n−k)!P(n,\,k) = \frac{n!}{(n-k)!}

Combinations — number of selections of kk items from nn, where order doesn’t matter:

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

Decision tree: does the order matter? If yes, use a permutation. If no, use a combination. Awarding gold/silver/bronze → permutation. Picking a 3-person committee → combination.

Worked example 1 — permutation

From 8 runners, how many ways to award gold, silver, and bronze? Order matters (1st ≠ 2nd ≠ 3rd):

P(8, 3)=8!5!=8⋅7⋅6=336P(8,\,3) = \frac{8!}{5!} = 8 \cdot 7 \cdot 6 = 336

Worked example 2 — combination

From 10 students, how many ways to choose a 3-person committee? Order doesn’t matter:

C(10, 3)=10!3!⋅7!=10⋅9⋅83⋅2⋅1=120\begin{aligned} C(10,\,3) &= \frac{10!}{3! \cdot 7!} \\ &= \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} \\ &= 120 \end{aligned}

How to type your answer

A single integer. Examples: 120, 336, 792.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

5!=?5! = ?

Problem 2

P(6, 2)=?P(6,\,2) = ?

Problem 3

C(5, 2)=?C(5,\,2) = ?

Problem 4

C(6, 3)=?C(6,\,3) = ?

Practice

Standard problems matching the lesson.

Problem 5

4!=?4! = ?

Problem 6

7!=?7! = ?

Problem 7

P(5, 3)=?P(5,\,3) = ?

Problem 8

P(8, 2)=?P(8,\,2) = ?

Problem 9

C(7, 2)=?C(7,\,2) = ?

Problem 10

C(8, 3)=?C(8,\,3) = ?

Problem 11

Arrangements of 4 books on a shelf\text{Arrangements of 4 books on a shelf}

Problem 12

Choose a 3-person committee from 10 students\text{Choose a 3-person committee from 10 students}

Problem 13

Award gold/silver/bronze among 8 runners\text{Award gold/silver/bronze among 8 runners}

Problem 14

4-letter codes from A,B,C,D,E,F (no repeats)\text{4-letter codes from A,B,C,D,E,F (no repeats)}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

C(10, 4)=?C(10,\,4) = ?

Problem 16

P(10, 4)=?P(10,\,4) = ?

Problem 17

9!=?9! = ?

Problem 18

Arrangements of the letters M, A, T, H\text{Arrangements of the letters M, A, T, H}

Problem 19

Seat 5 people in a row of 5\text{Seat 5 people in a row of 5}

Problem 20

C(12, 5)=?C(12,\,5) = ?

Problem 21

Choose 3 from a class of 20\text{Choose 3 from a class of 20}

Problem 22

Choose 4 books from 15 for vacation\text{Choose 4 books from 15 for vacation}