Number Theory

Combination with Repetition Calculator

Analyze prime factors, divisibility congruences, modular arithmetic, and integer properties for Combination with Repetition.

Reviewed by Sanjay Samanta & Academic Review Board
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Last updated: August 2026
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Verified Mathematical Solution

Result:

Number of combinations with repetition:

Understanding Combinations with Repetition

Combinations with repetition, also known as combinations with replacement, determine the number of ways to choose r items from a set of n distinct items, where repetition of items is allowed.

The formula for combinations with repetition is given by:

$$C(n+r-1, r) = \binom{n+r-1}{r} = \frac{(n+r-1)!}{r!(n-1)!}$$

Example:

Suppose you want to choose 2 scoops of ice cream (r=2) from 3 available flavors: vanilla, chocolate, and strawberry (n=3), and you can choose the same flavor twice. The possible combinations are:

  • Vanilla, Vanilla
  • Vanilla, Chocolate
  • Vanilla, Strawberry
  • Chocolate, Chocolate
  • Chocolate, Strawberry
  • Strawberry, Strawberry

Using the calculator with n=3 and r=2, you will find there are 6 combinations, which matches our example.

Direct Answer & Overview
Verified Educational Guide

How to Calculate Combination with Repetition

Analyze prime factors, divisibility congruences, modular arithmetic, and integer properties for Combination with Repetition.

Primary Mathematical Formula Standard Mathematical Model
Standard Equation
ƒ(x)
Q.E.D.
C(n,r)=(nr)=n!r!(n−r)!C(n, r) = \binom{n}{r} = \frac{n!}{r!(n - r)!}
Evaluated with exact mathematical formulation • Rigorously verified
Exact Formula
Input Parameters
Required
1
Total number of distinct items (n): Value for Total number of distinct items (n)
2
Number of items to choose (r): Value for Number of items to choose (r)
Expected Outputs
Calculated
Computed Combination with Repetition Calculator result with step-by-step mathematical breakdown
Exact numerical value and verified formula evaluation
Worked Numerical Example
Instant Verification
Calculate the result for Combination with Repetition given the input parameter values: Total number of distinct items (n) = 0, Number of items to choose (r) = 0.
→ Identify and verify the provided inputs (Total number of distinct items (n) = 0, Number of items to choose (r) = 0). Ensure units and signs are standardized before calculating.; Substitute the values into the mathematical relation: C(n, r) = \binom{n}{r} = \frac{n!}{r!(n - r)!}.
Result verified and calculated via Combination with Repetition Calculator

What Is the Combination with Repetition Calculator?

Analyze prime factors, divisibility congruences, modular arithmetic, and integer properties for Combination with Repetition.

What are Combinations with Repetition?

Combinations with repetition are a way to count the number of selections of items from a set where you are allowed to choose the same item multiple times. Unlike regular combinations where each item can be chosen at most once, combinations with repetition allow for items to be repeated in a selection.

This concept is useful in various scenarios, such as when you are selecting items from a menu where you can order the same dish multiple times, or when distributing identical items into distinct containers.

Formula and Calculation

The formula to calculate combinations with repetition is given by C(n+r-1, r), where n is the number of types of items to choose from, and r is the number of items to choose. This formula can be expanded as:

$$C(n+r-1, r) = \frac{(n+r-1)!}{r!(n-1)!}$$

This calculator simplifies the process of computing this value by directly applying this formula. Simply input the total number of distinct items (n) and the number of items you wish to choose (r), and the calculator will provide the result.

Uses and Applications

  • Inventory Management: Determining the number of ways to stock shelves with different types of products, allowing for multiple units of each product.
  • Menu Planning: Calculating meal combinations when you can choose multiple servings of the same dish.
  • Probability Problems: Solving problems where outcomes can be repeated, such as in certain types of sampling.
  • Computer Science: In scenarios like distributing identical tasks to different processors.

How to Use the Combination with Repetition Calculator

Using this calculator is straightforward. Enter your known values into the fields below, and the solver will compute the result immediately:

• Total number of distinct items (n)

Example input: 0.

• Number of items to choose (r)

Example input: 0.

Formula Reference
\(C(n, r) = \binom{n}{r} = \frac{n!}{r!(n - r)!}\)

Worked Example: Step-by-Step Combination with Repetition Problem

Worked Example
Problem Statement

Calculate the result for Combination with Repetition given the input parameter values: Total number of distinct items (n) = 0, Number of items to choose (r) = 0.

1

Collect and Verify Input Parameters

Identify and verify the provided inputs (Total number of distinct items (n) = 0, Number of items to choose (r) = 0). Ensure units and signs are standardized before calculating.

2

Substitute Values into the Governing Formula

Substitute the values into the mathematical relation: C(n, r) = \binom{n}{r} = \frac{n!}{r!(n - r)!}.

C(n, r) = \binom{n}{r} = \frac{n!}{r!(n - r)!}
3

Perform Step-by-Step Arithmetic Evaluation

Evaluate all operations following the strict mathematical order of operations (PEMDAS/BODMAS).

4

Format and Validate the Output

Round the final calculated numerical value to the required precision and verify against boundary conditions.

Final Result Result verified and calculated via Combination with Repetition Calculator

How to Calculate Combination with Repetition Step-by-Step

Understanding the underlying solution workflow helps build mathematical intuition and independently verify results:

1
Identify and note down the given values for: Total number of distinct items (n), Number of items to choose (r).
2
Set up the primary formula: \(C(n, r) = \binom{n}{r} = \frac{n!}{r!(n - r)!}\). Substitute the identified values into their respective positions.
3
Perform prime factorization, division, or modulo arithmetic to solve the number theory problem.
4
Round the final calculated answer to the required decimal accuracy or significant figures.

Real-World Applications of Combination with Repetition Calculator

Practical scenarios where combination with repetition calculator calculations are applied across engineering, business, and everyday problem solving:

Actuarial Insurance Risk Pricing

Actuaries model mortality tables, extreme weather events, and claim probabilities to establish sustainable policy premiums and cash reserves.

Cryptographic Key Security Combinatorics

Cybersecurity specialists compute permutation spaces to ensure encryption keys cannot be brute-forced within practical time limits.

Quality Assurance Acceptance Sampling

Inspectors use hypergeometric and binomial probability models to accept or reject massive shipment lots based on small random test samples.

Common Pitfalls & Mistakes to Avoid

Key calculation errors to avoid when computing combination with repetition calculator:

Confusing Permutations (Order Matters) with Combinations (Order Irrelevant)

Use permutations nPr when sequence order is significant (e.g. lock combinations, podium finishes). Use combinations nCr when selecting an unordered committee or subset.

Multiplying Probabilities of Dependent Events Without Conditional Adjustment

P(A and B) = P(A) · P(B) applies only to independent events. For dependent events, you must use P(A and B) = P(A) · P(B|A).

Adding Probabilities Without Subtracting Joint Intersection (Double Counting)

By the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B). Only omit the joint term if events are mutually exclusive (disjoint).

Key Terminology Glossary

Essential terms and definitions related to combination with repetition calculator:

Total number of distinct items (n) The Total number of distinct items (n) input parameter for the Combination with Repetition Calculator. Enter numerical values to execute calculations.
Number of items to choose (r) The Number of items to choose (r) input parameter for the Combination with Repetition Calculator. Enter numerical values to execute calculations.
Sample Space The comprehensive set of all possible outcomes resulting from an idealized random experiment.
Combinations vs Permutations Combinations count unordered subsets (nCr); permutations count ordered arrangements (nPr).
Verified STEM Methodology

About the Combination with Repetition Calculator

The Combination with Repetition Calculator is maintained by Basic Math Tools, an educational platform committed to providing accurate STEM and financial computing tools. Every tool processes calculations transparently in your browser for privacy, instant responsiveness, and mathematical accuracy.

If you have suggestions or questions regarding mathematical formulas, please review our Editorial Policy or contact our math team.

Fact-Checked & Verified • Computational Accuracy Standards
Updated August 2026 • Editorial Policy
Authored By
Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By
Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

Found an error or have an improvement suggestion? Report a calculation issue

Frequently Asked Questions

What is the difference between permutations and combinations?
The essential distinction is order: in permutations (nPr = n! / (n - r)!), order matters (e.g., assigning 1st, 2nd, and 3rd place prizes, or setting a lock passcode). In combinations (nCr = n! / (r!(n - r)!)), order does not matter (e.g., selecting a committee of 3 people, or dealing a hand of playing cards). For the same n and r, permutations always equal or exceed combinations.
What are independent events versus mutually exclusive events?
Mutually exclusive (disjoint) events cannot happen simultaneously (P(A and B) = 0; e.g., rolling a 2 and rolling a 5 on a single die roll). Independent events are events where the occurrence of one does not affect the probability of the other (P(A and B) = P(A) × P(B); e.g., flipping heads on a coin and rolling a 6 on a die).
How does conditional probability P(A|B) relate to Bayes' Theorem?
Conditional probability P(A|B) is the probability that event A occurs given that event B has already occurred: P(A|B) = P(A ∩ B) / P(B). Bayes' Theorem reverses this perspective: P(A|B) = [P(B|A) × P(A)] / P(B), allowing you to update the probability of a hypothesis (prior) in light of new observed evidence.
What is the Binomial Probability formula and when is it applicable?
The Binomial distribution P(X = k) = (n choose k) p^k (1 - p)^{n - k} applies when an experiment satisfies four conditions: fixed number of trials n, only two possible outcomes per trial (success or failure), constant probability of success p across all trials, and mutually independent trials.
What is expected value E(X) in probability modeling?
Expected value is the long-run theoretical average outcome of a random variable over repeated trials: E(X) = Σ [x_i · P(x_i)]. In games of chance or financial investments, a positive expected value indicates a profitable long-term proposition, while a negative expected value reflects a mathematical disadvantage.

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