Atotal of 32 students have entered a spelling contest there are 6 medals for first through sixth place that will be awarded in how many different ways can the medals be awarded?

Here the order matters (the medal of the 1st is different from the medal of the 2nd, etc.)
So it's a permutation (arrangement) of 6 chosen among 31:

 ³²P₆ = (n!)/(n-r)! = 32!/26! = 652,458,240

Another method:
the 1st medal could be won by 32
the  2nd medal could be won by 31 .
the  3dd medal could be won by 30
.etc. etc.
the  6th medal could be won by 27
.
Then the number of possibilities = 32x31x30x29x28x27 =652,458,240