We have an integer array with unique elements and we want to do the permutations that have an element fixed, in other words, these permutations should have a certain element at the same position than the original.

These permutations will be called: **permutations with one fixed point**.

Let's see an example with an array of four elements and we want the permutations that have a coincidence **only at  index 0**, so these permutations are (the permutations between parenthesis):
``` 
arr = [1, 2, 3, 4]
      (1, 3, 4, 2)
      (1, 4, 2, 3)
Two permutations matching with arr only at index 0
``` 

Let's see the permutations of the same array with only one coincidence at index **1**:
``` 
arr = [1, 2, 3, 4]
      (3, 2, 4, 1)
      (4, 2, 1, 3)
Two permutations matching with arr only at index 1
```
Once again, let's see the permutations of the same array with only one coincidence at index **2**:
``` 
arr = [1, 2, 3, 4]
      (2, 4, 3, 1)
      (4, 1, 3, 2)
Two permutations matching with arr only at index 2
```
Finally, let's see the permutations of the same array with only one coincidence at index **3**:
``` 
arr = [1, 2, 3, 4]
      (2, 3, 1, 4)
      (3, 1, 2, 4)
Two permutations matching with arr only at index 3
```
For this array given above (arr) :

- We conclude that we have 8 permutations with one fixed point (two at each index of arr).

- We may do the same development for our array, `arr`, with two fixed points and we will get `6` permutations.

- There are no permutations with coincidences only at three indexes.

- It's good to know that the amount of permutations with no coincidences at all are `9`. See the kata Shuffle It Up!!

In general:

- When the amount of fixed points is equal to the array length, there is only one permutation, the original array.

- When the amount of fixed points surpasses the length of the array, obvously, there are no permutations at all.

Create a function that receives the length of the array and the number of fixed points and may output the total amount of permutations for these constraints.

Features of the random tests:
```
length of the array = l
number of fixed points = k
10 ≤ k ≤ l ≤ 9000
```
See the example tests!
Enjoy it!!

Ruby versin will be released soon.

#Note: This kata was published previously but in a version not well optimized.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'from functools import lru_cache\n\n@lru_cache(maxsize=None)\ndef f(n):\n    if n <= 2:\n        return n - 1\n    return (n-1) * (f(n-1) + f(n-2))\n\ndef fixed_points_perms(n,k):\n    if n == k:\n        return 1\n    elif k >= n - 1:\n        return 0\n    elif k == 0:\n        return f(n)\n    return n * fixed_points_perms(n-1, k-1) // k'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'fixed_points_perms'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 5668