Permutations

Simple: learn it → see it → practice it.

LEARN EXAMPLES PRACTICE
LEARN

What is a permutation?

Order matters. Changing the order gives a new result.
Example: 321 is not the same as 123.

When do you use permutations?

  • Different places or prizes (like 1st/2nd/3rd)
  • Assigning special jobs or roles
  • Anytime the order makes a new answer

What do n and r mean?

n = total choices you have

Example: 8 people in a race → n = 8

r = number of spots to fill

Example: 3 medal places → r = 3

How do you find the answer?

Super simple:
Start with n.
Multiply by one less each time, do this r times.
(No repeats allowed!)
It's just: n Ă— (n-1) Ă— (n-2) ... (stop after r numbers)

Mini example

How many ways can 3 people get 1st, 2nd, and 3rd?
3 Ă— 2 Ă— 1 = 6
Quick tip: How do I know if it's a permutation?

If the roles or order matter, it's a permutation.

EXAMPLES

Example 1: Podium

Before you start:
n = how many choices you have at the start. Here: 12 runners.
r = how many spots you are filling. Here: 3 (gold, silver, bronze).

Question: How many ways can gold, silver, and bronze medals be given out among 12 runners?

Why does order matter?
Gold, silver, and bronze are different places—giving Alice gold and Bob silver is NOT the same as giving Bob gold and Alice silver.

Math: Start at 12 and multiply down 3 times: 12 · 11 · 10 = 1320

Why this is a permutation: The order you give out the medals changes who wins what, so each arrangement is unique.

Example 2: Roles

Before you start:
n = how many people you can pick from. Here: 25 students.
r = how many different roles you are assigning. Here: 3 (President, Vice-President, Treasurer).

Question: In how many ways can you assign President, Vice-President, and Treasurer from 25 students?

Why does order matter?
Each role is different. Assigning Sam as President and Maria as Treasurer is NOT the same as Maria as President and Sam as Treasurer.

Math: Start at 25 and multiply down 3 times: 25 · 24 · 23 = 13,800

Why this is a permutation: The order of the roles changes the result. Each assignment gives different students different jobs.

Example 3: Not a permutation

Before you start:
n = total students to pick from. Here: 18 students (for example).
r = number to choose for the committee. Here: 5.

Question: How many ways can you choose 5 students to be on a committee?

Not a permutation. On a committee, the order does NOT matter—just who is in the group. Order doesn’t create a new outcome.
This is a combination problem, not a permutation.

Quick calculator

Type n and r. It shows the “multiply down” steps.

P(10,4) = 5040
10 · 9 · 8 · 7
PRACTICE

Permutation Check

Is it a permutation?
  • Roles or spots are different (e.g., President vs. Secretary)
  • Order matters for the outcome (e.g., first place ≠ second place)
  • Each arrangement creates a new result

Exam-Style Permutation Practice

Try each group. Click "Show solution" to check your work.

1. People / Lineups

Q1. In how many ways can 5 students line up for lunch from a class of 24?
Show solution n = 24 because there are 24 students to choose from.
r = 5 because we are arranging 5 in a line.
Answer: 24 Ă— 23 Ă— 22 Ă— 21 Ă— 20 = 5,100,480
Q2. In how many ways can President, Vice-President, and Secretary be chosen from 12 students?
Show solution n = 12 because there are 12 students eligible.
r = 3 because there are 3 different positions.
Answer: 12 Ă— 11 Ă— 10 = 1,320
Q3. In how many ways can gold, silver, and bronze medals be awarded among 10 runners?
Show solution n = 10 because there are 10 runners.
r = 3 because there are 3 distinct medals.
Answer: 10 Ă— 9 Ă— 8 = 720

2. Letters / Digits

Q4. How many ways can you make a 4-letter code using the letters A, B, C, D, E, F if each letter is used only once?
Show solution n = 6 because there are 6 letters to choose from.
r = 4 because the code is 4 letters long.
Answer: 6 Ă— 5 Ă— 4 Ă— 3 = 360
Q5. You have 8 different digits (0-7). How many different 3-digit numbers can you form if no digit repeats?
Show solution n = 8 because there are 8 digits available.
r = 3 because the number has 3 digits.
Answer: 8 Ă— 7 Ă— 6 = 336
Q6. How many 5-letter “words” can you form from the word PIRATES if each letter is used at most once per word?
Show solution n = 7 because "PIRATES" has 7 distinct letters.
r = 5 because each "word" uses 5 letters.
Answer: 7 Ă— 6 Ă— 5 Ă— 4 Ă— 3 = 2,520

3. Restrictions

Q7. In how many ways can 6 people stand in a line if Ali must be first?
Show solution n = 5 because after Ali (already placed first), 5 people are left.
r = 5 because there are 5 spots left to fill.
Answer: 5 Ă— 4 Ă— 3 Ă— 2 Ă— 1 = 120
Q8. How many ways can 5 books be arranged on a shelf if Book A and Book B must be together?
Show solution Treat A & B as a single unit: now 4 “items” to arrange (= 4! = 24 ways). A & B can switch order: 2 ways.
n = 5 books, r = 5 spots.
So: 4! Ă— 2 = 24 Ă— 2 = 48
Q9. How many ways can 7 people arrange themselves in a row if Sara and Mia cannot be next to each other?
Show solution First, total arrangements: 7! = 5040.
Number where Sara and Mia are together: treat as a “block”: 6! × 2 = 720 × 2 = 1440.
Subtract: 5040 – 1440 = 3,600
n = 7 people, r = 7 spots.

4. With Repeated Items / Harder Restrictions

Q10. How many ways can you order the letters in SUCCESS?
Show solution The word SUCCESS has 7 letters: S(3), U(1), C(2), E(1).
n = 7 total letters.
r = 7 because all are arranged.
Answer: 7! / (3! Ă— 2! Ă— 1! Ă— 1!) = 5040 / (6Ă—2) = 5040 / 12 = 420
Q11. In how many ways can the letters of the word LEVEL be arranged?
Show solution LEVEL: 5 letters, L(2), E(2), V(1)
n = 5 total letters.
r = 5 since all letters are arranged.
Answer: 5! / (2! Ă— 2! Ă— 1!) = 120 / 4 = 30
Q12. How many ways can the letters of the word BANANA be arranged?
Show solution BANANA has 6 letters: A(3), N(2), B(1)
n = 6 total letters.
r = 6, arranging all.
Answer: 6! / (3! Ă— 2! Ă— 1!) = 720 / (6Ă—2) = 720 / 12 = 60