Calculate ordered arrangements with repetition (n^r). Supports variable pool sizes per position, position breakdown, entropy, and reference tables.
Permutations with repetition count the ordered arrangements when items can be reused. For r positions each with n choices, the total is n^r (or n₁ × n₂ × ... × nᵣ when positions have different pool sizes). This is the multiplication principle — the most fundamental counting rule in combinatorics.
This calculator handles both simple cases (same n for all positions) and general cases (different pool sizes per position, like license plate formats). It computes the total count, entropy bits, probability of any specific outcome, and provides a position-by-position breakdown.
Real-world applications include counting PINs (10^4 = 10,000), license plate numbers (26³ × 10⁴ = 175,760,000), dice outcomes (6^r), computer passwords, combination locks, and any scenario where you fill ordered slots from discrete option sets. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results.
While n^r is simple for equal pool sizes, real-world counting often involves positions with different numbers of options (letters+digits in license plates, mixed-radix systems). This calculator handles both cases, breaks down the count position by position, and provides the reference table for quick comparison. Keep these notes focused on your operational context.
Simple case (same pool): Total = n^r General case (different pools): Total = n₁ × n₂ × ... × nᵣ This is the multiplication principle (product rule). Entropy = log₂(Total) bits Examples: 4-digit PIN: 10^4 = 10,000 License AAA-1234: 26³ × 10⁴ = 175,760,000 3d6 dice: 6³ = 216
Result: 6 × 6 × 6 = 216
Rolling three six-sided dice gives 6³ = 216 possible outcomes. Each outcome (like 3-5-2) is equally likely with probability 1/216 ≈ 0.46%. The order matters: (1,2,3) and (3,2,1) are different outcomes.
The multiplication principle (product rule) states: if a process has k stages with n₁, n₂, ..., nₖ options respectively, the total outcomes are n₁ × n₂ × ... × nₖ. It's the basis for counting everything from passwords to genetic sequences (4^L for DNA of length L) to machine configurations.
n^r grows exponentially with r, which has profound implications for security (large password spaces), data encoding (bits per symbol), and computational complexity (search spaces). This growth is both a blessing (cryptographic security) and a curse (combinatorial explosion in optimization problems).
In information theory, the entropy of a random variable drawn uniformly from n^r outcomes is r × log₂(n) bits. This is the amount of information needed to specify one particular outcome. It connects counting (combinatorics) to communication (how many bits to transmit) and security (how hard to guess).
With repetition: each position can use any item (n^r total). Without repetition: once an item is used, it's removed from the pool (P(n,r) = n!/(n−r)!). A PIN allows repeats (1111 is valid); a race finishing order does not.
10^4 = 10,000 PINs (0000 through 9999). Each digit has 10 choices, and digits can repeat. Security relies on lockout policies, not combinatorial difficulty.
Because you multiply the number of choices at each step. If you choose a shirt (5 options), pants (3 options), and shoes (4 options), total outfits = 5 × 3 × 4 = 60.
A base-n number with r digits can represent exactly n^r values. Binary (n=2): 8 bits = 2^8 = 256 values. Decimal (n=10): 3 digits = 10^3 = 1,000 values. The counting is identical.
License plates (letters + digits), phone numbers (area code + subscriber), file naming conventions, and many real-world coding systems have positions with different character sets. Use this as a practical reminder before finalizing the result.
Exponentially in r. 2^10 = 1,024 but 2^20 = 1,048,576 and 2^30 = 1,073,741,824. Each additional position multiplies by n. This exponential growth makes brute-force attacks infeasible for large r.