Lesson 1
Counting basics
This lesson gives you the four tools that unlock a huge fraction of beginner combinatorics: the sum rule, the product rule, permutations, and combinations.
Your win for today
By the end, you should be able to look at a simple counting problem and decide, calmly, whether you should add, multiply, arrange, or choose.
1. Two master questions
Are these separate cases?
If you are counting one valid outcome from several non-overlapping buckets, you usually add.
Are these stages of one process?
If you make one choice and then another, you usually multiply.
2. Worked example, sum rule
How many ways can you choose a dessert if the menu offers 4 cakes or 3 ice creams, and you choose exactly one dessert?
- The cases are disjoint: cake or ice cream.
- You choose one dessert, not one of each.
- So the count is 4 + 3 = 7.
3. Worked example, product rule
How many 3-character codes can be made from 2 letters followed by 1 digit, if letters can be A, B, or C and digits can be 0 through 9?
- First letter: 3 choices.
- Second letter: 3 choices.
- Digit: 10 choices.
- Stages multiply, so 3 · 3 · 10 = 90.
4. Order matters vs order does not matter
Permutations
If you line things up, assign ranks, or care who sits where, order matters. That is a permutation problem.
P(n,r) = n! / (n-r)!
Combinations
If you just pick a group and swapping names changes nothing, order does not matter. That is a combination problem.
C(n,r) = n! / (r!(n-r)!)
5. Worked example, same numbers, different meaning
Suppose 5 students are available.
- Choose a president and vice president: order matters, so P(5,2) = 5 · 4 = 20.
- Choose a 2-person committee: order does not matter, so C(5,2) = 10.
Same people, same number chosen, different answer because the structure of the task changed.
6. Tiny retrieval drill
- Cover the formulas.
- Say out loud: add for cases, multiply for stages, permute for order, combine for groups.
- Wait a few minutes, then write the four ideas from memory.
This slight friction is useful. It builds storage strength, not just recognition.
Primary source for this lesson
Read the counting section in Discrete Mathematics: An Open Introduction for a clear textbook treatment, then dip into the MIT notes when you want a denser reference.
Ask follow-up questions
If any example felt shaky, ask me about it. I can generate more practice, explain why a method fits, or build the next lesson from where you got stuck.