In how many outcomes can we get 5 balls in 10 balls in any order?

Question

In order to enter the Lottery, you choose five different numbers in the range 1 to 53, and write them, in an order of your choice, on an entry form. ''' You win Prize 3 if your five numbers occur anywhere within the ten-number selection made by the machine (including those positions for which you win Prize 2), and in any order. In how many of the outcomes of part (a) do you win Prize 3? '''

So our teacher has told us to do:

1. fix 5 balls (1 way)
2. 48C5 to choose another 5 balls.
3. Mix them all up to the first 5 fixed balls can occur anywhere (10!)

so we do 48C5*10!.

My question is why do we use combinatorics and NOT permutations? since on the previous example : ''' You win Prize 2 if the first five balls selected by the machine match the five numbers you chose, and are arranged in any order (including that for which you win Prize ''' we used Fix 5 balls.

choose another 5 (48P5)