【Example 8】 How many terms are there in the expansion of an $n$-order determinant?


Since the expansion of a determinant is the algebraic sum of the products of all elements from different rows and different columns, each term in the expansion has the form $a_{1_{1}} a_{2 i_{i}} \cdots a_{n_{i}}$, and $i_{1}, i_{2}, \cdots, i_{n}$ exactly take all different permutations of $1,2, \cdots, n$. Therefore, by the counting formula for permutations, there are $A_{n}^{n}=n$! different terms in the expansion: