Permutations with Repetition Calculator

About the Permutations with Repetition Calculator

Permutations with repetition count ordered arrangements when each position can reuse the available options. For r positions with n choices each, the total is n^r. If the positions have different pool sizes, multiply them together: n₁ × n₂ × ... × nᵣ.

This calculator handles both the equal-pool case and mixed formats such as license plates, product codes, or any system where different positions draw from different character sets. It reports the total count, probability of one exact outcome, entropy in bits, and a position-by-position breakdown.

Typical use cases include PINs, passwords, dice rolls, serial-number formats, and any other ordered slot-filling problem where repetition is allowed.

When This Page Helps

The simple n^r formula covers many examples, but real counting problems often mix letters, digits, symbols, or category-specific slots. Breaking the total into per-position factors makes it easier to verify a format and explain where the size of the search space comes from.

How to Use the Inputs

1. Choose simple mode (same pool per position) or general mode (different pool per position).
2. In simple mode, enter n (choices per position) and r (number of positions).
3. In general mode, enter comma-separated pool sizes (e.g., 26,26,26,10,10,10,10).
4. Or click a preset for license plates, dice, locks, etc.
5. Review the total permutations, entropy, and probability.
6. Examine the position breakdown table.
7. Use the reference table for nearby n^r values.

Example Calculation

Tips & Best Practices

• This is the product rule: fill position 1 (n₁ ways), then position 2 (n₂ ways), so total = n₁ × n₂ × ..
• n^r grows exponentially with r. Adding one position multiplies the total by n.
• For passwords, permutations with repetition (pool^length) is the correct model since characters can repeat.
• If positions have different option counts (like license plates), multiply each position's options together.
• Entropy in bits = log₂(total) gives a standardized measure of the "randomness" or "information content."
• Compare with permutations without repetition: n^r is always ≥ P(n,r) = n!/(n−r)! when r ≤ n.

The Multiplication Principle: Foundation of Counting

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.

Exponential Growth and Its Implications

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).

Connection to Information Theory

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).

Frequently Asked Questions

• What is the difference between permutations with and without repetition?

  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.

• How many 4-digit PINs are there?

  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.

• Why is this called the multiplication principle?

  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.

• How does this relate to number systems?

  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.

• When do positions have different pool sizes?

  License plates, serial numbers, phone numbers, product codes, and mixed-format IDs often give different positions different character sets. In those cases, multiply the allowed options for each position instead of using one shared n value.

• How fast does n^r grow?

  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.