Virginia SOL Math / Algebra II / Statistics

A2.ST.3

Permutations & Combinations

What you'll learn

  • Decide whether order matters in a counting problem
  • Compute permutations using nPr
  • Compute combinations using nCr

Words to know

Permutation —
an arrangement where order matters
Combination —
a selection where order does not matter
Factorial —
n! is the product n times ... times 1
nPr —
the number of ordered arrangements of r from n
nCr —
the number of unordered selections of r from n

Why it matters Counting arrangements and selections tells us how many passwords, teams, or seatings are possible, and whether order changes the count.

Worked examples

Model ItI do

From 5 students, how many ways to pick a president and a vice president?

Order matters here, since president and vice president are different roles, so use a permutation. There are 5 choices for president and 4 for vice president: 5P2 = 5 * 4 = 20 ways.

Try It TogetherWe do

From 5 students, how many ways to pick a committee of 2?

Order does not matter for a committee, so use a combination: 5C2 = (5 * 4) / (2 * 1) = 10 ways.

Now YouYou do

How many ways can 4 runners finish first and second?

Decide if order matters, then use the right formula. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.Does a race finish order (1st, 2nd, 3rd) use permutations or combinations?

  2. 2.Compute 5P2.

  3. 3.Compute 4! (four factorial).

On Level

The core Quick Check from the book.

  1. 1.Does a race finish order use permutations or combinations?

  2. 2.Does choosing a committee use permutations or combinations?

  3. 3.Compute 4P2.

  4. 4.Compute 4C2.

  5. 5.Compute 3! (three factorial).

Challenge

Stretch your thinking.

  1. 1.From 7 students, how many ways can a committee of 3 be chosen?

  2. 2.How many ways can 4 different books be arranged in a row on a shelf?

  3. 3.From 6 runners, how many ways can gold, silver, and bronze medals be awarded?