2 / 5
Counting and Combinatorics
Counting and Combinatorics
Counting principles organize how many ways an arrangement or selection can happen.
Counting Rules
Product rule: if one task has $m$ choices and the next has $n$ choices, then the pair has $mn$ outcomes.
Permutation: $$P(n,r)=\frac{n!}{(n-r)!}$$
Combination: $$\binom{n}{r}=\frac{n!}{r!(n-r)!}$$
Example 1
How many 3-letter arrangements can be formed from 5 distinct letters without repetition?
$P(5,3)=5\cdot4\cdot3=60$.
Example 2
How many 3-person committees can be chosen from 8 people?
$$\binom{8}{3}=56$$
Practice Problems
1. Use the product rule for 3 shirt choices and 2 pants choices.
2. Compute $4!$.
3. Compute $P(6,2)$.
4. Compute $\binom{5}{2}$.
5. When do you use combinations instead of permutations?
Show Answer Key
1. $6$ outfits
2. $24$
3. $30$
4. $10$
5. When order does not matter