# Integer derangements count

`integer-derangements-count` · version 1.0.0 · Numbers & math · free, no key needed

Compute !n, the number of derangements (permutations with no fixed point) of n items, exactly.

**Use when you need to: integer derangements count · subfactorial · permutations with no fixed points.**

## Supported

- integer derangements count
- subfactorial
- permutations with no fixed points

## Not supported

- negative index
- partial derangements
- floating-point approximation

## Behavior

- n is a JSON integer from 0 to 100000.
- !0 is 1 (the empty permutation) and !1 is 0 (a single item cannot avoid its own position).
- The sequence is computed via the exact integer recurrence !n = (n-1) * (!(n-1) + !(n-2)), never floating point.
- Requests whose exact result would exceed 1000 digits are rejected rather than truncated.

## Input

- `n` (integer, required): min 0; max 100000

## Output

- `value` (string, required)

## Limits

- max n: 100000
- max digits: 1000

## Example

Request input:

```json
{
  "n": 4
}
```

Response:

```json
{
  "result": {
    "value": "9"
  }
}
```

## How to call it

### MCP

Connect `https://computefirst.net/mcp` ([setup](/docs#connect)), then call `execute` with:

```json
{
  "id": "integer-derangements-count",
  "version": "1.0.0",
  "input": {
    "n": 4
  }
}
```

### HTTP (no key)

```sh
curl -X POST https://computefirst.net/v1/tools/integer-derangements-count/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"n":4}'
```

The machine-readable contract is at [/v1/tools/integer-derangements-count/versions/1.0.0](/v1/tools/integer-derangements-count/versions/1.0.0).

### CLI

```sh
node cli.mjs run integer-derangements-count 1.0.0 --input input.json --base-url https://computefirst.net
```

Get the client at [/clients/cli/](/clients/cli/).

## Related tools

- [Integer permutations count](/tools/integer-permutations-count): Compute the exact number of k-permutations of n distinct items, P(n, k) = n! / (n-k)!.
- [Integer binomial coefficient](/tools/integer-binomial-coefficient): Compute the exact binomial coefficient C(n, k), the number of ways to choose k items from n.
- [Integer divmod](/tools/integer-divmod): Integer quotient and remainder truncated toward zero.
- [Integer totient](/tools/integer-totient): Compute Euler's totient function: the count of integers in [1, value] coprime to value.
- [Array integer adjacent diff](/tools/array-integer-adjacent-diff): Replace an array of canonical integer strings with consecutive first differences.
- [Array integer clamp](/tools/array-integer-clamp): Clamp an array of canonical integer strings to optional inclusive min and max bounds.