Power Rule

Power Rule

INFO

For any set ( A ),

$$|\underset{n}{\underbrace{A\times A\times A\cdots \times A}}|=|A{|}^n$$

Uses

• To count the number of strings of length ( n ) over a finite alphabet ( A )
• To count the number of ( n )-length sequences using only the numbers:
  • $\{0,1,\dots ,|A|-1\}$
  • $\{1,2,\dots ,|A|\}$
  • Any other set with cardinality equal to $|A|$
• To count the number of ways of distributing $|A|$ distinct objects among $n$ people, possibly with the same person getting multiple objects

Example

How many 4-letter words can be formed over the Latin alphabet (26 letters)?

$$26^4$$

How many 8-character passwords can be made using uppercase, lowercase, and digits?

$$(26+26+10{)}^8=62^8$$

TIP

The Power Rule assumes independence between choices — each position in the sequence can be filled without constraints from the others.