# Multinomial Coefficient & Probability Calculator

> Multinomial Coefficient & Probability Calculator is a paid API for AI agents from probability-calc.x402tookit.com, paid per call via x402, $0.02/call, status unknown (last checked 2026-10-29).

Computes the multinomial coefficient (number of arrangements of items with repeats, e.g. MISSISSIPPI) and optionally the multinomial probability given outcome probabilities.

## Facts

- Endpoint: GET https://probability-calc.x402tookit.com/multinomial?utm_source=zero.xyz
- Price: $0.02/call
- Payment: x402
- Status: unknown
- Last checked: 2026-10-29
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/multinomial-coefficient-probability-calculator-ac00b891
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_4yUEDUI1_WTGrriRXMpbT

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-probability-calculator-ac00b891
```

Example prompt: How many distinct ways can the letters in MISSISSIPPI be arranged? Also give me the multinomial probability if the letter probabilities are 0.1, 0.4, 0.4, 0.1.

## When to prefer this

Use this endpoint when you need to count distinct arrangements of a multiset (e.g. letters in a word with repeats) or compute the exact multinomial probability for a set of outcome counts given per-outcome probabilities. Prefer this over a general combinatorics endpoint when your problem maps directly to arrangements with repeated items or multinomial distributions.

## Known failure modes

- Missing both 'word' and 'counts' parameters returns an error
- Probabilities that don't sum to 1.0 may produce an error or invalid result
- Non-integer or negative group sizes may be rejected
- Empty or single-character word returns a trivial coefficient of 1
- Very large inputs may exceed numeric precision, returned in scientific notation

## How this service works

Multinomial coefficient (arrangements of repeated items like MISSISSIPPI) and multinomial probability for given odds.

## Output

Returns the multinomial coefficient (as a string, possibly in scientific notation), total number of items, the group sizes used, the formula string, a human-readable summary, and optionally the multinomial probability if outcome probabilities were supplied. Also includes metadata like source and an asOf timestamp.

## 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"
   ],
   "properties": {
    "type": {
     "type": "string",
     "const": "http"
    },
    "method": {
     "enum": [
      "GET"
     ],
     "type": "string"
    },
    "queryParams": {
     "type": "object",
     "required": [],
     "properties": {
      "word": {
       "type": "string",
       "description": "Or a word/string: the repeated letters set the group sizes (MISSISSIPPI)"
      },
      "probs": {
       "type": "string",
       "description": "Optional outcome probabilities, comma separated, one per count, summing to 1: gives the multinomial probability of exactly these counts"
      },
      "counts": {
       "type": "string",
       "description": "Group sizes, comma separated: 1,4,4,2 (give counts or word)"
      }
     }
    }
   },
   "additionalProperties": false
  },
  "output": {
   "type": "object",
   "required": [
    "type"
   ],
   "properties": {
    "type": {
     "type": "string"
    },
    "example": {
     "type": "object",
     "properties": {
      "asOf": {
       "type": [
        "null",
        "string"
       ]
      },
      "total": {
       "type": [
        "null",
        "number"
       ]
      },
      "digits": {
       "type": [
        "null",
        "number"
       ]
      },
      "source": {
       "type": [
        "null",
        "string"
       ]
      },
      "formula": {
       "type": [
        "null",
        "string"
       ]
      },
      "letters": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "string"
       }
      },
      "reasons": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "string"
       }
      },
      "summary": {
       "type": [
        "null",
        "string"
       ]
      },
      "verdict": {
       "type": [
        "null",
        "string"
       ]
      },
      "degraded": {
       "type": [
        "boolean",
        "null"
       ]
      },
      "disclaimer": {
       "type": [
        "null",
        "string"
       ]
      },
      "groupSizes": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "number"
       }
      },
      "notChecked": {
       "type": [
        "array",
        "null"
       ],
       "items": {
… (truncated)
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/multinomial-coefficient-probability-calculator-ac00b891/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from probability-calc.x402tookit.com](https://www.zero.xyz/host/probability-calc.x402tookit.com/llms.txt)