How to implement falling factorial in Python
Falling factorial is a notion used in counting issues to describe the product of decreasing integers up to a given point. It is especially helpful when selecting a certain number of elements from a bigger collection is required, and order is important.
Definition of falling factorial
The sum of the first
Generalization of falling factorial
The falling factorial, which we can use to select the top
Recurrence relation
The recurrence connection can be used to relate the falling factorial to the factorial:
By using this relation, we may create a recursive definition of the falling factorial by expressing it in terms of a smaller falling factorial.
Implementation
Let’s implement the falling factorial in Python:
import mathdef falling_factorial(n, k):if n < k:raise ValueError("n must be greater than or equal to k")result = math.factorial(n) // math.factorial(n - k)return result# Example usagen = 10k = 3result = falling_factorial(n, k)print(f"The falling factorial ({n})_{k} is: {result}")
Explanation
Line 1: Here, we import the
mathmodule.Lines 3–9: We calculate the falling factorial of a given number
nwithkterms.Lines 5–6: We check if the value of
n(representing the base) is less than the value ofk(representing the number of terms). Ifnis less thank, it means there aren't enough terms to calculate the falling factorial, so it raises aValueErrorwith an appropriate message.Line 8: We calculate the falling factorial using the formula
n! / (n - k)!. It first calculates the factorial ofnusingmath.factorial(n)and then divides it by the factorial ofn - kto get the falling factorial. The//operator is used for integer division to ensure the result is an integer.Line 9: It returns the calculated falling factorial value.
Lines 12–15: We print the result is an example of using the function with
n=10andk=3.
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