The equasion has only one solution.
Each medal is different and is handed out depending on the outcome of the race/order of the racers, so we're counting the number of permutations. 4 medals are given to 4 runners, so there are

possible ways to do it.
Since there are a total of 48 beads and each bead will either be blue or yellow, in order to find out how many of each colour bead she will use, do the following:
Divide 48 by 3 to get 16 yellow beads.
Now that you know how many yellow beads there are, just subtract that from 48:
48 - 16 = 32
Therefore, she will use 16 yellow beads and 32 blue beads.