Permutation with Repetition Calculator
Calculate permutations where elements can be repeated using n^r formula
Calculate Permutations with Repetition
Number of distinct object types available
Number of positions to fill (repetition allowed)
Number of Permutations
Formula
P(n,r) with repetition = n^r
Where n = 10 (types of objects) and r = 4 (positions to fill)
Each of the 4 positions can be filled with any of the 10 objects
Comparison with Other Methods
Example Calculation
4-Digit PIN Problem
Problem: How many 4-digit PINs can be created using digits 0-9?
Given: n = 10 digits (0,1,2,3,4,5,6,7,8,9), r = 4 positions
Note: Repetition is allowed (PIN like 1122 or 0000 is valid)
Solution
P(10,4) with repetition = 10^4
P(10,4) = 10 × 10 × 10 × 10
P(10,4) = 10,000 possible PINs
This includes PINs like 0000, 1234, 9999, 2468, etc.
Key Concepts
With Repetition
Each element can be used multiple times
Formula: n^r
Order Matters
ABC ≠ ACB ≠ BAC
Different arrangements = different results
Independent Choices
Each position has n choices
Total: n × n × ... × n (r times)
Quick Examples
Mathematical Properties
n^0 = 1 (empty arrangement)
n^1 = n (single position)
n^r ≥ P(n,r) without repetition
Exponential growth with r
Understanding Permutations with Repetition
What are Permutations with Repetition?
Permutations with repetition occur when you arrange items from a set where each item can be used multiple times. Unlike permutations without repetition, the same element can appear in multiple positions. The formula is simply n^r, where n is the number of distinct objects and r is the number of positions.
Key Characteristics
- •Repetition allowed: Same element can be used multiple times
- •Order matters: ABC ≠ BCA ≠ CAB
- •Independent choices: Each position has n options
- •Simple formula: n^r (n to the power of r)
Common Applications
- •PIN numbers and passwords
- •License plate combinations
- •Product codes and serial numbers
- •Lottery number selections
- •DNA sequence combinations
- •Multi-choice question answers
Memory Tip: Think "Each position = n choices, multiply r times!" Since repetition is allowed, every position independently has all n options available.
Mathematical Formula
Permutations with repetition allowed
Where:
- n: Number of distinct object types
- r: Number of positions to fill
- n^r: n multiplied by itself r times
- Result: Total number of arrangements
Why n^r?
- Position 1: n choices
- Position 2: n choices (repetition allowed)
- Position 3: n choices
- Total: n × n × n × ... = n^r
Example: Password Creation
Problem: How many 3-character passwords using letters A, B, C?
Solution: 3^3 = 27 passwords (AAA, AAB, AAC, ABA, ABC, etc.)
Example: Dice Rolls
Problem: How many outcomes when rolling a die 3 times?
Solution: 6^3 = 216 outcomes (111, 112, 113, ..., 666)