If one stage has m options and the next has n, the two together have m × n outcomes.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Factorials
n! multiplies every whole number from n down to 1, and counts the arrangements of n distinct items.
Permutations
A permutation counts arrangements, where order matters. nPr = n! ÷ (n − r)!.
Combinations
A combination counts selections, where order does not matter. nCr = n! ÷ (r!(n − r)!).
The link
A combination is a permutation with the r! internal orderings divided out, so nCr is always the smaller.
Why probability needs this
A theoretical probability is favourable outcomes over total outcomes, and counting is how both numbers are found.
The multiplication principle
If one choice has m options and a second has n, there are mn combined outcomes. Almost all counting reduces to this, applied repeatedly, and drawing a small tree confirms it.
Permutations count arrangements
Order matters, so choosing 3 from 10 in order gives 10 × 9 × 8. First, second and third place are different outcomes even with the same three people.
Combinations count selections
Order does not matter, so divide the permutation count by the number of ways the chosen items could be arranged. A committee of three is one outcome however the names are listed.
Deciding which applies
Ask whether swapping two chosen items gives a different outcome. If yes, it is a permutation; if no, a combination. That single question settles it more reliably than any keyword.
Step 2: Try It Yourself
Tap and try it out.
Arrangements
Selections
Arrangements has the most. It has 10 more than Selections.
Step 3: Watch an Example
One step at a time.
Watch Rosa Count a Committee
Rosa must choose 3 people from 6 for a committee.
- Step 1
Swapping two members gives the same committee, so order does not matter.
Step 4: Your Turn
Practice makes it stick.
The Arrangement
Problem 1 of 2
How many ways can 4 books be arranged in a row?
The Selection
Problem 2 of 2
How many ways to choose 2 toppings from 5?
Count the Ways
1 of 8
What is 5!?
2 of 8
What is 6C2?
3 of 8
What is 5P2?
4 of 8
3 starters and 4 mains. How many two-course meals?
5 of 8
How many 3-digit codes from digits 0 to 9, repeats allowed?
6 of 8
What is 7C7?
7 of 8
Sort each situation by whether order matters.
Tap something to move it.
- Empty
- Empty
8 of 8
What is 4C2?
Step 5: Quick Check
Show what you know.
Question 1 of 2
How many ways can 3 books be arranged in a row?
Question 2 of 2
Choosing a committee of 3 from 10 uses which count?
What You Learned
- Multiply the options at each stage to count outcomes.
- Permutations count arrangements; combinations count selections.
- nCr is nPr with the r! orderings divided out.