1. The problem is to understand the expansion of $n!$ (n factorial). 2. The notation $n!$ means the product of all positive integers from 1 up to $n$. 3. Expressed mathematically: $$n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$$ 4. For example, if $n=5$, then: $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$ 5. There is no simpler algebraic expansion of factorial since it is a product, but it can be defined recursively: $$n! = n \times (n-1)!$$ with the base case $$0! = 1$$ 6. Factorials grow very quickly and are used widely in combinatorics, probability, and algebra. Final answer: The expansion of $n!$ is simply the product $$n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$$.