Integrated Math 2 / Unit 9: Probability
9.2 Counting Principles
Notes
Counting methods avoid listing every outcome.
- Multiplication principle: if one choice has $m$ ways and another has $n$ ways, together there are $m \cdot n$ ways.
- Permutations count ordered arrangements: $\displaystyle _nP_r = \frac{n!}{(n - r)!}$.
- Combinations count groups where order does not matter: $\displaystyle _nC_r = \frac{n!}{r!(n - r)!}$.
- Example: choose 3 people from 6 for a committee: $\displaystyle _6C_3 = \frac{6 \cdot 5 \cdot 4}{3 \cdot 2 \cdot 1} = 20$.
Practice
A store has 4 shirts and 3 pairs of pants. How many different outfits are possible?
Multiply: $4 \cdot 3 = 12$.
Five runners race. In how many ways can first, second and third place be awarded?
Order matters: $5 \cdot 4 \cdot 3 = 60$.
A club of 6 students picks 3 to form a committee with no roles. How many committees are possible?
Order does not matter, so $\displaystyle _6C_3 = \frac{6 \cdot 5 \cdot 4}{3 \cdot 2 \cdot 1} = 20$.
Ready for more? Take the Unit 9 test.