Permutations with Repetition
Ordered arrangements of r items from n when repetition is allowed.
Formula
Variables
Example
3-digit PIN: 10³ = 1000
How it works
This formula counts how many ordered outcomes you can make when each of the r positions has n possible choices and the same choice can be used again. The key idea is the multiplication rule: n choices for the first slot, n for the second, and so on, giving n × n × ... × n = n^r. It applies because each slot is independent of the others, so repetitions do not reduce the number of possibilities.
When to use it
Use this when order matters, you have a fixed number of positions, and repeated choices are allowed. Common examples include passwords, codes, repeated draws with replacement, and simple counting of sequences.
Worked example, step by step
Example: How many 4-letter codes can be made using the letters A, B, C, and D if letters may repeat? Step 1: Number of choices per slot = 4 Step 2: Number of slots = 4 Step 3: Apply the formula: 4^4 Step 4: Calculate: 4 × 4 × 4 × 4 = 256 Final answer: 256 possible codes
Common mistakes to avoid
- Using nPr or nCr even though repetition is allowed and order matters
- Forgetting that each position has the same number of choices, so the total is a power
- Confusing this with problems where choices cannot repeat, which need a different formula
Did You Know?
This is why a 4-digit PIN has exactly 10,000 possibilities.
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More in Probability & Combinatorics
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Factorial
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