Permutations with Repetition – When Repeats Are Allowed
Ordinary permutations assume each item can only be used once. But many real situations allow repetition – a PIN code can reuse the same digit twice, and a password can reuse the same letter. When repetition is allowed and order still matters, the count is simply nr: n choices for each of r positions, all multiplied together.
This formula is really just the Fundamental Counting Principle again, applied to r identical stages that each have exactly n options – a reminder that nearly everything in this section traces back to that one core idea.
The Repetition Formula
The number of ordered sequences of length r, chosen from n options with repetition allowed, is nr.
How many 4-digit PINs are possible, using digits 0-9 with repeats allowed?
104 = 10,000 possible PINs.
Real-Life Application
- PIN codes and passwords: most codes allow repeated characters.
- DNA sequences: counting possible sequences from 4 repeatable bases (A, C, G, T).
- Dice rolls: counting possible results from rolling several dice in a row.
Key Takeaways
- Permutations with repetition count ordered sequences where items can repeat.
- The formula is nr, for r positions each with n options.
- This is the Fundamental Counting Principle applied to r identical stages.
Practice: Permutations with Repetition
Permutations with Repetition
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