Are these separate cases?
If you are counting one valid outcome from several non-overlapping buckets, you usually add.
Lesson 1, Counting basics
Counting basics
Section titled “Counting basics”The two master questions
Section titled “The two master questions”Are these stages of one process?
If you make one choice and then another, you usually multiply.
Worked example, sum rule
Section titled “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.
Quick check. If there are 5 teas or 4 coffees and you choose exactly one drink, how many choices are there?
Answer: 9.
Worked example, product rule
Section titled “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.
Quick check. If you have 4 shirts and 6 hats, how many shirt-hat outfits can you make?
Answer: 24.
Order matters vs order does not matter
Section titled “Order matters vs order does not matter”Permutations
If you line things up, assign ranks, or care who sits where, order matters.
Use P(n, r) = n! / (n-r)!.
Combinations
If you just pick a group and swapping names changes nothing, order does not matter.
Use C(n, r) = n! / (r!(n-r)!).
Same numbers, different meaning
Section titled “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.
Quick check. From 5 players, how many 3-player teams can you choose?
Answer: 10.
Tiny retrieval drill
Section titled “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.
That little bit of friction is useful. It builds storage strength, not just recognition.
Primary source for this lesson
Section titled “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
Section titled “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.
Next: Counting cheat sheet