Add generalized nCr (combinations) calculator for real n and integer r (#13623)

* Create ncr_combinations.py

* Update ncr_combinations.py

* Update ncr_combinations.py

* Update ncr_combinations.py

* Update ncr_combinations.py

* updating DIRECTORY.md

---------

Co-authored-by: Christian Clauss <cclauss@me.com>
Co-authored-by: cclauss <cclauss@users.noreply.github.com>

Author: Naman-Vasudev

DIRECTORY.md

@@ -815,6 +815,7 @@
   * [Modular Exponential](maths/modular_exponential.py)
   * [Monte Carlo](maths/monte_carlo.py)
   * [Monte Carlo Dice](maths/monte_carlo_dice.py)
+  * [Ncr Combinations](maths/ncr_combinations.py)
   * [Number Of Digits](maths/number_of_digits.py)
   * Numerical Analysis
     * [Adams Bashforth](maths/numerical_analysis/adams_bashforth.py)

maths/ncr_combinations.py

@@ -0,0 +1,77 @@
+"""
+Generalized combinations (n choose r) calculator for real total and integer choose.
+Wikipedia URL:
+https://en.wikipedia.org/wiki/Binomial_coefficient
+"""
+
+from math import factorial as math_factorial
+
+
+def combinations(total: float, choose: int) -> float:
+    """
+    Compute the number of combinations using the formula:
+
+        combinations = total * (total-1) * ... * (total-choose+1) / choose!
+
+    Parameters
+    ----------
+    total : float
+        Total number of items. Can be any real number.
+    choose : int
+        Number of items to select. Must be a non-negative integer.
+
+    Returns
+    -------
+    float
+        The number of combinations.
+
+    Raises
+    ------
+    ValueError
+        If choose is not a non-negative integer.
+
+    Examples
+    --------
+    >>> combinations(5, 2)
+    10.0
+    >>> combinations(5.5, 2)
+    12.375
+    >>> combinations(10, 0)
+    1.0
+    >>> combinations(0, 0)
+    1.0
+    >>> combinations(5, -1)
+    Traceback (most recent call last):
+        ...
+    ValueError: choose must be a non-negative integer
+    >>> combinations(5, 2.5)
+    Traceback (most recent call last):
+        ...
+    ValueError: choose must be a non-negative integer
+    """
+    if not isinstance(choose, int) or choose < 0:
+        raise ValueError("choose must be a non-negative integer")
+
+    if choose == 0:
+        return 1.0
+
+    numerator = 1.0
+    for i in range(choose):
+        numerator *= total - i
+
+    denominator = math_factorial(choose)
+    return numerator / denominator
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    # Example usage
+    total_input = float(input("Enter total (real number): ").strip() or 0)
+    choose_input = int(input("Enter choose (non-negative integer): ").strip() or 0)
+    print(
+        f"combinations({total_input}, {choose_input}) = "
+        f"{combinations(total_input, choose_input)}"
+    )