Question

a total 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?

Answer 1

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