Combinations With Repetition Calculator - Multiset Coefficients and Stars and Bars

A combinations with repetition calculator is a combinatorics tool that returns the multiset coefficient (n multichoose r) for any n and r. Type the number of distinct categories and the number of items to choose, and the calculator returns the exact integer (n + r - 1)! / (r! x (n - 1)!) using BigInt arithmetic.

• • Pizza Toppings and Menu Combos: Count the multisets of r toppings chosen from n options, for example 3 toppings from 4 options gives 20 multisets when repeats are allowed.
• • Dice Rolls and Repeated Trials: Count multisets of dice outcomes, for example the 56 distinct triples of three dice.
• • Stars and Bars Distributions: Count the ways to distribute r identical objects into n distinct boxes.
• • Loot Drops and Inventory Picks: Count item multisets when an item type can be chosen more than once, for example rarity tiers in a loot table.

The multiset coefficient is also written as (n multichoose r) or the binomial coefficient C(n + r - 1 choose r), and it is the natural answer whenever the same category can be picked again. The count applies to repeated sampling, identical ball draws from a labeled urn, and any selection where order does not matter but repeats do.

When the experiment allows repeats, the combination calculator is the general tool that returns the standard n choose r value with a repetition toggle for cross-checking, so the two counts can be compared on the same inputs.

The calculator applies the multiset coefficient formula, evaluates the binomial coefficient (n + r - 1 choose r) with BigInt arithmetic, and returns the result alongside the factorial form and a standard C(n, r) cross-check so the difference between no-repetition and with-repetition counts is visible on the same page.

• n: Number of distinct categories or types, an integer from 1 to 50. n = 1 returns 1 regardless of r because every multiset from one type is identical.
• r: Number of items to choose, an integer from 0 to 25. r = 0 returns 1 by convention because there is one way to choose nothing.
• (n + r - 1): Top of the binomial coefficient in the equivalent standard C(n + r - 1, r) form. For n = 4 and r = 3 the top is 6 and the coefficient is C(6, 3).
• (n multichoose r): The exact integer multiset coefficient, computed with BigInt so it stays precise for inputs like (50 multichoose 25).

The factorial form and the standard C(n, r) cross-check appear alongside the result, so the difference between with-repetition and no-repetition counts is visible on the same page. Stars and bars give the same number: distributing r identical items into n distinct boxes has (n + r - 1 choose r) solutions.

According to Omni Calculator - Combinations with Repetition, the combinations-with-repetition formula is (n + r - 1)! / (r! x (n - 1)!), and choosing 3 toppings from 4 pizza options gives 20 multisets.

According to Wolfram MathWorld - Multiset, the number of multisets of size r chosen from a set of n distinct types is (n multichoose r) = (n + r - 1)! / (r! x (n - 1)!).

Because the multiset coefficient is built from (n + r - 1)!, r!, and (n - 1)!, the factorial calculator is the right companion when the user wants to read off each factorial value one at a time before plugging them into the ratio.

Four ideas make the multiset coefficient easier to apply, and each one maps to a real input, output, or rule the calculator exposes:

These ideas prevent the most common mistakes: forgetting the + r - 1 in the top of the binomial, using the standard C(n, r) when the problem allows repeats, and confusing unordered multiset counts with ordered outcomes. The multichoose result feeds straight into the ordered count P = (n multichoose r) x r!, and the permutation and combination calculator returns both the no-repetition and with-repetition counts on the same inputs.

Follow these four steps to compute the multiset coefficient for any n and r with the combinations with repetition calculator:

1. 1 Enter the Number of Distinct Categories: Type the number of distinct categories or types, for example 4 pizza toppings, 6 die faces, or 13 card ranks. The calculator accepts n from 1 to 50.
2. 2 Enter the Number of Items to Choose: Type r, the number of items to choose, for example 3 toppings, 3 dice, or 5 cards. The calculator accepts r from 0 to 25 and returns 1 when r = 0.
3. 3 Read the Multisets Result: Check the (n multichoose r) value in the result panel. Results up to 15 digits use thousands separators; larger counts switch to scientific notation.
4. 4 Verify With the Formula and Cross-Check: Use the factorial form, the stars and bars interpretation, and the standard C(n, r) cross-check to confirm the result by hand.

For example, with n = 4 and r = 3 the calculator shows (4 multichoose 3) = 20 and standard C(4, 3) = 4, matching the pizza topping case where 3 toppings from 4 options with repetition allowed give 20 multisets but only 4 unordered sets without repetition.

Once the multiset coefficient is known, the probability calculator divides favorable outcomes by the total multisets to return the matching event probability for dice, coins, and repeated sampling problems.

Using a dedicated combinations with repetition calculator has several practical advantages over computing (n multichoose r) by hand:

• • Exact BigInt Multiset Coefficient: BigInt arithmetic keeps the result the exact integer, not a rounded floating-point approximation, even for inputs like (50 multichoose 25).
• • Readable Factorial Form: The formula field shows the multiplication in factorial form, for example 6! / (3! x 3!) = 20, so the user can verify the calculation step by step.
• • Stars and Bars Interpretation: The formula string links the result to the stars and bars model, so the same number answers distribution and selection problems.
• • Standard C(n, r) Cross-Check: The cross-check field returns C(n, r) on the same n and r, so the with-repetition count can be compared to the no-repetition count.
• • Wide Input Range: The calculator accepts n from 1 to 50 and r from 0 to 25, covering the most common multiset problems from pizza toppings to inventory picks.
• • Reset for the Next Problem: Press Reset to restore the default n = 4 and r = 3, useful for tutors and students working through several multiset problems in a row.

Most introductory multiset problems in a statistics or probability unit are answered in seconds once the user stops expanding (n + r - 1 choose r) by hand, and the same multiset count feeds straight into dice probability, loot drop sizing, and stars and bars distributions.

When the experiment has more than one independent stage and each stage is its own multiset problem, the fundamental counting principle calculator multiplies the per-stage multiset counts to size the full sample space.

A few real-world factors change how the multiset coefficient applies and what the calculator returns:

• • The calculator assumes each category is distinct and the items within a category are identical. If the items within a category are also distinct, switch to a permutation formula or use the product of binomial coefficients for each category.
• • It does not apply to problems with restrictions such as a maximum number of items per box. For those cases, apply inclusion-exclusion or use a generating function alongside the basic multiset count.

The same (n + r - 1 choose r) value appears in the stars and bars theorem, the multiset coefficient, and the no-restriction distribution count, which is why a single (n multichoose r) result drives so many introductory combinatorics problems.

According to Wikipedia - Multiset, the number of multisets of size r drawn from a set of n distinct elements is the multiset coefficient (n + r - 1 choose r), and the stars and bars theorem counts distributions of r identical items into n distinct boxes.