Falling and rising factorials

Falling and rising factorials are two mathematical concepts often used in combinatorics and algebra to describe specific products of sequences of numbers. They are particularly useful in the context of permutations, combinations, and polynomial expansions. Here's an overview of both: ### Falling Factorials The falling factorial, denoted as \( (n)_k \), is defined as the product of \( k \) consecutive decreasing integers starting from \( n \).