Advanced Counting Techniques
Advanced counting techniques use the multiplication principle, permutations, and combinations to count arrangements and selections in complex probability and combinatorics problems.
What You Need to Know
Key Concept Diagram
The multiplication principle: if event A can occur in m ways and event B in n ways, both can occur in m x n ways
A permutation P(n,r) = n!/(n-r)! counts ordered arrangements of r items from n
A combination C(n,r) = n!/(r!(n-r)!) counts unordered selections of r items from n
Use permutations when order matters and combinations when order does not matter
n! (n factorial) = n x (n-1) x (n-2) x ... x 2 x 1
Key Vocabulary
Permutation
An ordered arrangement of a set of items
Combination
An unordered selection of items from a set
Factorial
The product of all positive integers up to a given number; n! = n x (n-1) x ... x 1
Multiplication principle
If one event can happen in m ways and a second in n ways, both together can happen in m x n ways
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
How many ways can 5 students be arranged in a line?
Question 2
How many ways can 3 students be chosen from a group of 8 (order does not matter)?
Question 3
A PIN uses 4 different digits from 0-9. How many different PINs are possible?
Key Concepts Summary
- ●The multiplication principle: if event A can occur in m ways and event B in n ways, both can occur in m x n ways
- ●A permutation P(n,r) = n!/(n-r)! counts ordered arrangements of r items from n
- ●A combination C(n,r) = n!/(r!(n-r)!) counts unordered selections of r items from n
- ●Use permutations when order matters and combinations when order does not matter
- ●n! (n factorial) = n x (n-1) x (n-2) x ... x 2 x 1