This is a simple counting principle problem.
125 people can possibly come in 1st.
Once one is the winner, only 124 can come in 2nd
And now only 123 remain to come in 3rd
Your total is simply 125*124*123=1,906,500 different ways.
This is also known as a permutation, where order matters. You can use the formula for permutations to get the same answer:
125P3 = 125!/(125-3)!