Added Derangements Calculator (#9927)

* Added Derangement calculator

* Updated derangement.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Updated derangement.py

* updating DIRECTORY.md

* Apply suggestion from @cclauss

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>
Co-authored-by: cclauss <cclauss@users.noreply.github.com>

Author: 3 people

DIRECTORY.md

@@ -726,6 +726,7 @@
   * [Continued Fraction](maths/continued_fraction.py)
   * [Decimal Isolate](maths/decimal_isolate.py)
   * [Decimal To Fraction](maths/decimal_to_fraction.py)
+  * [Derangement](maths/derangement.py)
   * [Dodecahedron](maths/dodecahedron.py)
   * [Double Factorial](maths/double_factorial.py)
   * [Dual Number Automatic Differentiation](maths/dual_number_automatic_differentiation.py)
@@ -876,6 +877,7 @@
   * [Two Pointer](maths/two_pointer.py)
   * [Two Sum](maths/two_sum.py)
   * [Volume](maths/volume.py)
+  * [Weighted Average](maths/weighted_average.py)
   * [Zellers Congruence](maths/zellers_congruence.py)
 
 ## [Matrix](matrix)

maths/derangement.py

@@ -0,0 +1,42 @@
+"""
+A Python implementation for finding number of
+derangements possible for k objects
+https://en.wikipedia.org/wiki/Derangement
+"""
+
+
+def derangement(objects: int) -> int:
+    """
+    Calculates the number of derangements of k objects.
+    :param objects:the number of objects ( -1 < objects < 1560 )
+    :return :the number of derangements
+    :raises :ValueError: If objects is negative.
+
+    Examples:
+    >>> derangement(3)
+    2
+    >>> derangement(5)
+    44
+    >>> derangement(10)
+    1334961
+    """
+    if objects < 0:
+        raise ValueError("k must be a non-negative integer. Retry")
+
+    # Base cases
+    if objects in (0, 1):
+        return 0
+
+    # Initialize the derangement counts
+    derange_1 = 1
+    derange_2 = 0
+    answer = 1
+
+    # Calculate derangements using dynamic programming
+    # Answer: F(n) = (n - 1) * ( F(n - 1) + F(n - 2) )
+    for i in range(3, objects + 1):
+        answer = (i - 1) * (derange_1 + derange_2)
+        derange_2 = derange_1
+        derange_1 = answer
+
+    return answer