Free Combinations with Repetition Calculator

Combinations with Repetition: An Overview

Combinations with repetition—also called combinations with replacement or multiset selections—form a foundational counting technique in combinatorics. They quantify the number of ways to choose a sample of size $r$ from a set of $n$ distinct categories, where each category may be selected more than once. This concept arises in many practical scenarios: assessing lottery odds, analyzing hash collisions in computer science, or determining the variety of possible portfolios when assets can be bought repeatedly.

Understanding the underlying Combination with Repetition Formula is valuable even when using an automated tool. The formula itself is compact:

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

where $n$ is the number of distinct objects, $r$ is the sample size, and the prime symbol distinguishes this from ordinary combinations (without repetition). The same quantity is sometimes written as $\overline{C}(n,r)$ or as the multiset coefficient $\left(\!\!\binom{n}{r}\!\!\right)$, and its derivation is often explained through the stars‑and‑bars method.

Worked Example: Five Digits from 0–9

Suppose you want to know how many combinations of five digits can be formed from the decimal digits 0 through 9 when repetition is permitted. Here $n = 10$ and $r = 5$. Substituting into the formula gives:

\begin{aligned} C'(10,5) &= \frac{(5 + 10 - 1)!}{5!\,(10 - 1)!} = \frac{14!}{5! \times 9!} \$$4pt] &= \frac{87,\!178,\!291,\!200}{120 \times 362,\!880} = 2002. \end{aligned}

Thus, there are 2002 distinct multisets (combinations) of five digits when repetition is allowed. If order were considered (permutations with repetition), the number would be $10^5 = 100,\!000$. The dramatic difference highlights the effect of ignoring order.

Using the Online Combinations with Repetition Calculator

The Combinations with Repetition Calculator (also known as a Combinations with Replacement Calculator or Multiset Combination Calculator) handles all the factorial arithmetic instantly. You simply enter the values for $n$ and $r$, and the tool returns $C'(n,r)$ without manual computation.

Importantly, this Combination with Repetition Online tool does more than just count: it acts as a combination generator with repetition, producing a readable list of every possible selection. This feature is especially useful for verification or when you need to inspect the combinations themselves rather than just their number.

Key Distinctions

To apply the results correctly, it is essential to differentiate combinations with repetition from related concepts:

• Combinations without repetition ($C(n,r) = \frac{n!}{r!(n-r)!}$) allow each object to be chosen at most once, yielding smaller counts for the same $n$ and $r$.
• Permutations with repetition ($n^r$) treat the order of selection as meaningful, which always gives larger counts than the corresponding combination case.

The calculator focuses on the “order does not matter, repetition allowed” scenario—a staple in probability problems, combinatorial design, and statistical sampling. By clarifying these distinctions, the tool serves as a reliable companion for both students and practitioners of discrete mathematics.