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AMC 10 & 12

Counting Principles

The addition, multiplication, and complement rules — three simple ideas that everything else in combinatorics builds on.

The multiplication principle

If a process has several independent, sequential steps, and step 1 can happen in mm ways, step 2 in nn ways (regardless of what happened in step 1), and so on, the total number of outcomes is the product:

m⋅n⋅⋯m \cdot n \cdot \cdots

The addition principle

If you're counting outcomes that split into cases, and those cases are genuinely disjoint (no outcome belongs to two cases at once), you add the counts:

∣A∪B∣=∣A∣+∣B∣(only when A∩B=∅)|A \cup B| = |A| + |B| \qquad\text{(only when } A \cap B = \varnothing\text{)}

The disjointness requirement is the whole trap here — if the cases can overlap, straight addition double-counts, and you need inclusion-exclusion instead (covered in a later lesson).

The complement principle

Sometimes it's much easier to count what you don't want than what you do. If the total number of outcomes is known, subtract the unwanted ones:

∣A∣=∣S∣−∣Ac∣|A| = |S| - |A^c|

This is the single highest-leverage trick in this lesson — any time a problem says “at least one” or asks you to exclude a messy condition, try complementing first.

Worked example

Problem: How many 3-digit numbers (100–999) contain at least one digit equal to 7?

Solution: Complementing is much easier than casework here. Total 3-digit numbers:999−100+1=900999-100+1=900. Count those with no 7's: the first digit has 8 choices (1–9, excluding 7), and each of the other two digits has 9 choices (0–9, excluding 7):

8×9×9=6488 \times 9 \times 9 = 648

So the count with at least one 7 is 900−648=252900 - 648 = 252.

Practice problems

1.A restaurant offers 4 appetizers, 6 entrées, and 3 desserts. How many different 3-course meals are possible?

2.How many outcomes are there when flipping 5 coins?

3.How many 4-digit numbers (1000–9999) contain no digit equal to 5?

4.Using the previous answer, how many 4-digit numbers contain at least one digit equal to 5?

Permutations & Combinations