01 Key Concepts
Multiplication Principle
If one task can be done in m ways and a second independent task in n ways, both together can be done in m*n ways.
Permutations
Arrangements where order matters. The number of ways to arrange r items from n distinct items is P(n,r) = n! / (n-r)!.
Combinations
Selections where order does NOT matter. The number of ways to choose r items from n distinct items is C(n,r) = n! / [r!(n-r)!].
Factorial Notation
n! (n factorial) means n*(n-1)*(n-2)*...*2*1. By definition, 0! = 1.
When to Use Permutations vs. Combinations
Use permutations when arrangement/order matters (like race rankings); use combinations when only the selection matters (like choosing a committee).
02 Key Formulas
- P(n,r) = n! / (n-r)!
- C(n,r) = n! / [r!(n-r)!]
03 Solved Examples
Example 1 How many ways can 3 runners finish 1st, 2nd, and 3rd out of 8 racers?
- Order matters here (1st is different from 2nd), so use permutations.
- P(8,3) = 8! / (8-3)! = 8! / 5! = 8*7*6.
Answer: 336 ways
Example 2 How many ways can a committee of 3 be chosen from 8 people?
- Order doesn't matter for a committee, so use combinations.
- C(8,3) = 8! / [3!*5!] = (8*7*6) / (3*2*1).
Answer: 56 ways
Example 3 A restaurant has 4 appetizers and 6 main courses. How many appetizer-main course combinations are possible?
- Use the multiplication principle: multiply the number of choices for each independent decision.
- 4 * 6.
Answer: 24 combinations
04 Practice Questions
1Find P(5,2).
20
2Find C(5,2).
10
3Find 5!.
120
4A password uses 2 letters followed by 3 digits; how many total combinations (using multiplication principle) if there are 26 letters and 10 digits, with repetition allowed)?
26*26*10*10*10 = 676,000
5When should you use combinations instead of permutations?
When the order of selection does not matter
📄 Combinatorics — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.