Definition: A partition of a set 𝑆 is a collection of subsets 𝑆1 , 𝑆2 , ⋯ , 𝑆𝑘 of 𝑆 such that each element of 𝑆
appears in exactly one the subsets:
𝑆 = 𝑆1 ∪ 𝑆2 ∪ ⋯ ∪ 𝑆𝑚
𝑆𝑖 ∩ 𝑆𝑗 = ∅, (𝑖 ≠ 𝑗)
Definition: A partition of a set 𝑆 is a collection of subsets 𝑆1 , 𝑆2 , ⋯ , 𝑆𝑘 of 𝑆 such that each element of 𝑆
appears in exactly one the subsets:
𝑆 = 𝑆1 ∪ 𝑆2 ∪ ⋯ ∪ 𝑆𝑚
𝑆𝑖 ∩ 𝑆𝑗 = ∅, (𝑖 ≠ 𝑗)
Addition Principle: Suppose that a set 𝑆 is partitioned into parts 𝑆1 , 𝑆2 , ⋯ , 𝑆𝑘 . The number of objects
in 𝑆 then equals the sum of the number of objects for each part.
|𝑆| = |𝑆1 | + |𝑆2 | + ⋯ + |𝑆𝑚 |
Many examples will incorporate this principle as part of the solution, but here are two using it alone:
Example 1:
Suppose we wanted to count how many courses were offered by Meredith College.
We could count the courses by department then add up all those numbers. However, this will only
give the correct answer if ___________(?)
Example 2:
Suppose you wanted to count the number of distinct courses you’ve taken. You could
simply count the number of courses taken each semester then add up all those numbers. However,
this will only give the correct answer if _________ (?)
Multiplication Principle: Suppose that a set 𝑆 consists of ordered pairs. In other words, 𝑠 ∈ 𝑆 implies
that 𝑠 = (𝑎, 𝑏) with 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵 for some sets 𝐴 and 𝐵. In this case:
|𝑆| = |𝐴| × |𝐵|
This is a fundamental principle which appears often and is quite useful. It actually can be derived
from the addition principle and the definition of multiplication as repeated addition.
Example 3:
A t-shirt is available as long sleeve vs. short sleeve, in four possible sizes and five
possible colors. How many distinct types of t-shirt are available?
Example 4:
How many two digit numbers have distinct digits, with both digits non-zero?
Multiplication Principle: Suppose that a set 𝑆 consists of ordered pairs. In other words, 𝑠 ∈ 𝑆 implies
that 𝑠 = (𝑎, 𝑏) with 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵 for some sets 𝐴 and 𝐵. In this case:
|𝑆| = |𝐴| × |𝐵|
This is a fundamental principle which appears often and is quite useful. It actually can be derived
from the addition principle and the definition of multiplication as repeated addition.
Example 5:
I need to create a class schedule that consists of five different courses. If there are 20
courses available, how many schedules are possible? Do we need to make any assumptions in order
to calculate our answer?
Example 6:
How many different 5-digit numbers can be constructed using the digits 1, 1, 1, 3, 8?
Example 7:
How many odd numbers between 1000 and 9999 have distinct digits?
NOTE: We’ve already seen two distinct types of problem – ones where the objects counted have a
particular order and ones where there is no ordering.
For each of these types, repetition may or may not be permitted.
More complex problems require a combining both the addition and the multiplication principles:
Example 8:
How many numbers between 1 and 10,000 have exactly one digit equal to 5?
Definition: An 𝑟-combination from a set 𝑆 is a selection of 𝑟 elements from 𝑆.
Definition: An 𝑟-permutation is from a set 𝑆 is an ordered sequence of 𝑟 elements from 𝑆.
Examples?
𝑛
Notation: ( ) is read as “𝑛 choose 𝑟” and it equals the number of ways of selecting an r-combination
𝑟
from a set with 𝑛 elements.