Factorial Notation
n! = n × (n−1) × (n−2) × ··· × 2 × 1
Read as "n factorial". Special case: 0! = 1
0!
1
1!
1
2!
2
3!
6
4!
24
5!
120
6!
720
7!
5,040
8!
40,320
9!
362,880
10!
3,628,800
Fundamental Counting Principle
If one event can occur in m ways and a second independent event can occur in n ways, the two events can occur together in m × n ways.
Total outcomes = n₁ × n₂ × n₃ × ···
Worked Example 1
A license plate has 3 letters followed by 4 digits. How many plates are possible?
Letters: 26 choices each × 3 positions = 26 × 26 × 26
Digits: 10 choices each × 4 positions = 10 × 10 × 10 × 10
Total = 26³ × 10⁴ = 17,576 × 10,000
= 175,760,000 possible plates
Permutations — Order Matters
A permutation is an ordered arrangement. P(n,r) = nPr counts the number of ways to arrange r items chosen from n distinct items.
P(n, r) = n! / (n − r)!
n = total items, r = items selected, order matters
Worked Example 2
How many ways can 8 runners finish 1st, 2nd, and 3rd?
P(8, 3) = 8! / (8−3)! = 8! / 5!
= (8 × 7 × 6 × 5!) / 5!
= 8 × 7 × 6
= 336 arrangements
Worked Example 3
How many 4-letter codes from the word MATH (no repeats)?
n = 4 letters, r = 4 positions
P(4, 4) = 4! / (4−4)! = 4! / 0!
= 24 / 1
= 24 arrangements
Permutations with Repeated Elements
When some items are identical, divide by the factorial of each repeated group:
n! / (n₁! × n₂! × ··· × nₖ!)
Example: Arrangements of MISSISSIPPI (11 letters: 1 M, 4 I, 4 S, 2 P) = 11! / (1! × 4! × 4! × 2!) = 34,650
Combinations — Order Doesn't Matter
A combination counts the number of ways to choose r items from n distinct items without regard to order.
C(n, r) = n! / (r! × (n − r)!)
Also written C(n,r), ⁿCᵣ, or {n choose r}
Worked Example 4
How many ways to choose 3 students from a class of 20 for a committee?
C(20, 3) = 20! / (3! × 17!)
= (20 × 19 × 18) / (3 × 2 × 1)
= 6,840 / 6
= 1,140 combinations
Worked Example 5
A 5-card hand from a 52-card deck — how many hands?
C(52, 5) = 52! / (5! × 47!)
= (52 × 51 × 50 × 49 × 48) / (5!)
= 311,875,200 / 120
= 2,598,960 hands
C(n,0) = 1
One way to choose nothing
C(n,n) = 1
One way to choose everything
C(n,r) = C(n, n−r)
Symmetry: choosing r is same as excluding n−r
Permutation vs Combination — Decision Guide
| Scenario | Order Matters? | Formula |
|---|---|---|
| Race finishing positions (1st, 2nd, 3rd) | Yes | P(n, r) |
| Seating arrangement at a table | Yes | P(n, r) or n! |
| Password / PIN digits | Yes | P(n, r) or nʳ with repetition |
| Committee selection (no roles) | No | C(n, r) |
| Lottery ticket (pick 6 of 49) | No | C(n, r) |
| Card hand from a deck | No | C(n, r) |
| Pizza toppings (choose 3 of 10) | No | C(n, r) |
| Arrange letters in a word | Yes | n! or n!/repeats |
Multi-Step Counting Problem
Worked Example 6
A club has 10 members. How many ways can they elect a president, VP, and a 3-person committee (with no person holding two roles)?
Step 1 — Elect president and VP (order matters)
P(10, 2) = 10 × 9 = 90
Step 2 — Choose 3 from remaining 8 (order doesn't matter)
C(8, 3) = 8!/(3!×5!) = 56
Step 3 — Multiply (Fundamental Counting Principle)
90 × 56 = 5,040
= 5,040 ways
Formula Quick Reference
Exam Strategy
Ask: does swapping matter?
If swapping two items gives a different valid outcome, use P(n,r). If swapping gives the same result, use C(n,r). "Committee of 3" → same; "1st, 2nd, 3rd place" → different.
Cancel factorials in the formula
In P(n,r) = n!/(n−r)!, the (n−r)! cancels with the bottom of n!, leaving just n×(n−1)×···×(n−r+1). Never compute the full factorial if you can cancel.
Break multi-stage problems apart
Identify each independent choice, compute P or C for each stage, then multiply them all together using the Fundamental Counting Principle.
Frequently Asked Questions
What is the difference between a permutation and a combination?
A permutation is an arrangement where order matters (1st, 2nd, 3rd are different outcomes). A combination is a selection where order does NOT matter (choosing 3 people for a committee — the order they're chosen is irrelevant). As a rule: if you can swap two elements and get a different outcome, it's a permutation.
When do I use nCr vs nPr on my calculator?
Use nPr (permutations) when arrangement matters — passwords, race finishing positions, seating arrangements. Use nCr (combinations) when only the group matters — lottery tickets, committee selections, card hands. nCr = nPr ÷ r! because combinations discard the r! orderings of each group.
What is 0! (zero factorial)?
0! = 1 by definition. This is required for the formulas to work correctly — for example, C(n,0) = n!/(0!·n!) = 1, meaning there is exactly one way to choose nothing from a set. The result 0! = 1 is also consistent with the recursive definition n! = n × (n−1)!.
Related Topics
Practice Counting Problems
Get step-by-step solutions for permutations, combinations, and multi-step counting problems — free on NailTheTest.
Start Practicing Free