# Multinomial Coefficient Calculator

> Multinomial Coefficient Calculator is a paid API for AI agents from combinatorics.openverbs.com, paid per call via x402, $0.004/call, status unknown (last checked 2026-09-14).

Computes the multinomial coefficient (k1+k2+…)! / (k1! k2! …) — the number of ways to partition n items into labelled groups of specified sizes.

## Facts

- Endpoint: POST https://combinatorics.openverbs.com/v1/multinomial
- Price: $0.004/call
- Payment: x402
- Status: unknown
- Last checked: 2026-09-14
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/multinomial-coefficient-calculator-c9993b88
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_3vxoxQF912W4HZ7YHvaGr

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability multinomial-coefficient-calculator-c9993b88 -d '<json body>'
```

Example prompt: How many ways can 12 people be divided into labelled groups of 4, 5, and 3? Calculate the multinomial coefficient for those group sizes.

## When to prefer this

Choose this endpoint when you need the exact multinomial coefficient for partitioning items into three or more labelled groups. For two-group partitions, the binomial coefficient endpoint on the same service may be simpler. For ordered arrangements, use the permutation endpoint. This endpoint handles sums up to 20000, making it suitable for large combinatorial problems that would overflow standard integer arithmetic.

## Known failure modes

- Sum of group sizes exceeds 20000 — returns an error
- Empty or missing groups array — validation error
- Non-integer or negative group sizes — schema rejection
- Exceeds 256 groups limit — rejected by maxItems constraint
- Network timeout on very large computations

## How this service works

Multinomial coefficient (k1+k2+…)! / (k1! k2! …) — the number of ways to partition n = Σki items into labelled groups of the given sizes. The sum must be ≤ 20000.

## Output

Returns the exact multinomial coefficient as a large integer (likely as a decimal string given the potential for very large values), representing the number of distinct ways to partition n = Σki items into the specified labelled groups.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "$schema": "https://json-schema.org/draft/2020-12/schema",
 "required": [
  "input"
 ],
 "properties": {
  "input": {
   "type": "object",
   "required": [
    "type",
    "method",
    "bodyType",
    "body"
   ],
   "properties": {
    "body": {
     "type": "object",
     "required": [
      "groups"
     ],
     "properties": {
      "groups": {
       "type": "array",
       "items": {
        "type": "string",
        "pattern": "^\\d+$"
       },
       "maxItems": 256,
       "minItems": 1,
       "description": "Group sizes as decimal strings; their sum must be ≤ 20000."
      }
     },
     "additionalProperties": false
    },
    "type": {
     "type": "string",
     "const": "http"
    },
    "method": {
     "enum": [
      "POST"
     ],
     "type": "string"
    },
    "bodyType": {
     "enum": [
      "json",
      "form-data",
      "text"
     ],
     "type": "string"
    }
   },
   "additionalProperties": false
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/multinomial-coefficient-calculator-c9993b88/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from combinatorics.openverbs.com](https://www.zero.xyz/host/combinatorics.openverbs.com/llms.txt)