Permutation & Combination Skill

Overview

The calculator takes two non-negative integers, `n` (the size of the set, up to 170) and `r` (how many are chosen). `r` larger than `n` is rejected with a 400 rather than returning a meaningless count, and so is anything that is not a whole number.

`permutation` is P(n,r) = n! / (n-r)!, the number of ordered arrangements of r items taken from n. It is computed as the product of the integers from n-r+1 up to n, so the (n-r)! term cancels instead of being built and divided. `combination` is C(n,r) = n! / (r! (n-r)!), obtained by dividing that product by r!. Both run in exact big-integer arithmetic, which is why C(30,15) returns 155117520 rather than a value a fraction below it.

The results are converted to JSON numbers on the way out. Counts up to 2^53 are exact; larger ones, which appear quickly as n grows, come back as the nearest double, correctly rounded from the exact value, so leading digits and magnitude are right but trailing digits are not. The cap at 170 exists because 171! is larger than a double can hold. Edge cases follow the usual conventions: r = 0 gives 1 for both, and r = n makes the permutation equal to n! and the combination 1.

`formulas` holds the two expressions as text and is null on the Free plan.

Live Test Permutation & Combination Skill Skill →

The tool

Once your client is connected to the VerveKit server, this appears in its tool list as PermutationCombinationSkill. It is read-only and open-world — it fetches and never mutates anything on your side — so most clients call it without asking you to confirm.

{
  "name": "PermutationCombinationSkill",
  "arguments": {
    "n": "10",
    "r": "3"
  }
}

You do not name the tool yourself; the model picks it. Asking about 10 in the terms this skill covers is enough for it to reach for PermutationCombinationSkill on its own — naming it explicitly also works, and is the way to force the call.

Connecting

One server URL covers every skill in the catalog, including this one. Authorization is OAuth: the client opens a browser once, and there is no key to paste into a config file.

{
  "mcpServers": {
    "vervekit": {
      "url": "https://api.vervekit.com/v1/mcp"
    }
  }
}

https://api.vervekit.com/v1/mcp

Per-client setup — Claude, Cursor, VS Code, ChatGPT — is on the MCP setup page.

Arguments

These are the properties on the tool's inputSchema, so a well-behaved client validates them before the call is made. Premium arguments are accepted on every plan but only take effect on plans that include them.

What the model gets back

The result carries a structuredContent object matching the tool's declared outputSchema, so a client reads fields without parsing prose. status is "ok" and error is null on success; a null field means the value was not available for that input, not that the call failed.

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{
  "status": "ok",
  "error": null,
  "data": {
    "n": 10,
    "r": 3,
    "permutation": 720,
    "combination": 120,
    "formulas": {
      "permutation": "P(n,r) = n! / (n-r)!",
      "combination": "C(n,r) = n! / (r! × (n-r)!)"
    }
  }
}

Response fields

Paths are relative to data. Premium fields are absent rather than zeroed on plans that do not include them, so check for presence instead of comparing to 0.

Failure modes

Errors come back as tool errors carrying a sentence the model can act on, not a bare status code. Error handling covers the full list.

A call costs 1 credit each time the tool actually runs; a model that reasons about the tool without calling it costs nothing.

Use cases

Education

Solve permutation and combination problems for homework and exams

Probability

Calculate odds and probabilities in statistics

Programming

Determine algorithm complexity and counting problems

Game Design

Calculate possible combinations for game mechanics

Other ways to use Permutation & Combination Skill

Set up Permutation & Combination Skill on VerveKit, or reach the same source a different way. Your VerveKit account and credits work on all of them — one key, one balance.

Related

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