Stars and bars API

Counts the ways to distribute identical items into distinct bins (with or without empty bins) via the stars-and-bars formula. Answers 'How many ways to split 10 identical balls into 4 boxes?', 'Non-negative integer solutions to x1+...+x4=10?'.

Price$0.04per request

MethodPOST

Route/v1/gametheory/stars-and-bars

StatusLive

MIME typeapplication/json

Rate limit120/minute

Cache0s public

combinatoricsstars-and-barscompositionsdistributionscountingballs-in-boxesdiscreteinteger-solutions

API URLhttps://x402.hexl.dev/v1/gametheory/stars-and-bars

Integration docs

Example request

{
  "items": 10,
  "bins": 4,
  "allowEmpty": true
}

Example response

{
  "items": 10,
  "bins": 4,
  "allowEmpty": true,
  "ways": 286,
  "waysExact": "286",
  "isExactInteger": true,
  "formula": "C(10 + 4 - 1, 4 - 1)",
  "interpretation": "Number of ways to distribute 10 identical items into 4 distinct bins (empty bins allowed): 286."
}

Input schema

{
  "type": "object",
  "required": [
    "items",
    "bins"
  ],
  "properties": {
    "items": {
      "type": "integer",
      "minimum": 0,
      "examples": [
        10
      ]
    },
    "bins": {
      "type": "integer",
      "minimum": 1,
      "examples": [
        4
      ]
    },
    "allowEmpty": {
      "type": "boolean",
      "examples": [
        true
      ]
    }
  }
}

Output schema

{
  "type": "object",
  "additionalProperties": true
}