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Discrete Math

Combinations with Replacement

Combinations with replacement. Free online Combinations with Replacement. Calculate combinations with replacement online — fast, accurate, mobile-friendly, no s

C(n+r-1, r)
35

Derivation

  1. ├── 01Givenn = 5, r = 3
  2. ├── 02Formulat};return n(t+a-1) / (n(a) × n(t-1))
  3. ├── 03Substitutet};return 5(t+a-1) / (5(a) × 5(t-1))
  4. └── 04Compute C(n+r-1, r)35
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§01What is

Understanding the Combinations with Replacement

The Combinations with Replacement computes C(n+r-1, r) from 2 inputs: n, r. Combinations with replacement.

Quick calculators for the math that shouldn’t need a notepad — instant, accurate, private to your browser. The Combinations with Replacement sits in that toolkit — it combinations with replacement. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

t};return n(t+a-1) / (n(a) × n(t-1))

Where

n
n
r
r
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: n = 5, r = 3.

  1. 01Start by noting the input — n: 5.
  2. 02Start by noting the input — r: 3.
  3. 03Substitute these values into the formula: t};return n(t+a-1) / (n(a) × n(t-1))
  4. 04Compute C(n+r-1, r): the calculator returns 35.
  5. 05Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Combinations with Replacement Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

n halved

n = 2.5 (from 5)

Keep every other input at its default and halve the n. See how c(n+r-1, r) responds.

  1. 01New n: 2.5
  2. 02Baseline C(n+r-1, r): 35
  3. 03New C(n+r-1, r): 4
  4. 04C(n+r-1, r) decreases by 88.6% → use this sensitivity to plan for real-world variation.
02 · PATTERN

n doubled

n = 10 (from 5)

Keep every other input at its default and double the n. See how c(n+r-1, r) responds.

  1. 01New n: 10
  2. 02Baseline C(n+r-1, r): 35
  3. 03New C(n+r-1, r): 220
  4. 04C(n+r-1, r) increases by 528.6% → use this sensitivity to plan for real-world variation.
03 · PATTERN

r halved

r = 1.5 (from 3)

Keep every other input at its default and halve the r. See how c(n+r-1, r) responds.

  1. 01New r: 1.5
  2. 02Baseline C(n+r-1, r): 35
  3. 03New C(n+r-1, r): 5
  4. 04C(n+r-1, r) decreases by 85.7% → use this sensitivity to plan for real-world variation.
04 · PATTERN

r doubled

r = 6 (from 3)

Keep every other input at its default and double the r. See how c(n+r-1, r) responds.

  1. 01New r: 6
  2. 02Baseline C(n+r-1, r): 35
  3. 03New C(n+r-1, r): 210
  4. 04C(n+r-1, r) increases by 500% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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