To solve this problem, we need to determine how many different ways we can award 10 different gift cards to 3,125 ticket buyers, assuming no one can win more than once.
This is a permutations problem, where we want to know the number of ways to choose and arrange 10 winners from a group of 3,125 people. The formula for permutations of [tex]\( n \)[/tex] items taken [tex]\( r \)[/tex] at a time is given by:
[tex]\[ P(n, r) = \frac{n!}{(n-r)!} \][/tex]
Here, [tex]\( n \)[/tex] is 3,125 (the number of ticket buyers), and [tex]\( r \)[/tex] is 10 (the number of gift cards). Plugging these values into the formula, we get:
[tex]\[ P(3125, 10) = \frac{3125!}{(3125-10)!} \][/tex]
Simplifying the expression inside the factorial, we have:
[tex]\[ \frac{3125!}{3115!} \][/tex]
This corresponds to one of the expression given in the choices:
[tex]\[ \frac{3,125!}{(3,125-10)!} \][/tex]
Therefore, the correct expression that represents the number of ways the gift cards can be awarded is:
[tex]\[ \boxed{\frac{3,125!}{(3,125-10)!}} \][/tex]
