Counting Principles & Permutations
Counting Principles & Permutations
If one task can be done in $m$ ways and a second independent task in $n$ ways, the pair can be done in $m \times n$ ways.
If task A can be done in $m$ ways and task B in $n$ ways, and the two tasks are mutually exclusive, then one or the other can be done in $m + n$ ways.
An ordered arrangement of $r$ objects chosen from $n$ distinct objects:
$$P(n,r) = \frac{n!}{(n-r)!}$$
$n! = n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1$, with $0! = 1$ by convention.
A restaurant offers 4 appetizers, 6 entrees, and 3 desserts. How many different meals are possible?
$$4 \times 6 \times 3 = 72 \text{ meals}$$
How many ways can 5 runners finish in 1st, 2nd, and 3rd place?
$$P(5,3) = \frac{5!}{2!} = \frac{120}{2} = 60$$
How many 4-digit codes can be formed from digits 0–9 with no repetition?
$$P(10,4) = 10 \cdot 9 \cdot 8 \cdot 7 = 5{,}040$$
How many ways can 7 people sit in a row?
$$7! = 5{,}040$$
Practice Problems
Show Answer Key
1. $10^3 = 1{,}000$
2. $720$
3. $8 \cdot 7 \cdot 6 = 336$
4. $P(10,3) = 720$
5. $26 \cdot 25 \cdot 24 = 15{,}600$
6. $P(12,4) = 11{,}880$
7. $26^3 \cdot 10^4 = 175{,}760{,}000$
8. $5! = 120$
9. $2^8 = 256$
10. $4! = 24$
11. $10 \cdot 9 \cdot 8 = 720$
12. $P(9,4) = 3{,}024$