Question

There are 10 students participating in a spelling bee. In how many ways can the students who go first and second be chosen?

Answer 1

90 different ways.
There are 10 ways to choose the first student that goes first, leaving 9 different ways a second student can be chosen. Using combinations, you multiply 10 by 9 to get a total of 90 different combinations, or ways students can be chosen to go first or second.

Answer 2

The order matters.

$k$ objects can be chosen out of $n$ objects, when the order matters, in $P(n,k)=\dfrac{n!}{(n-k)!}$ ways.

So, the answer is $P(10,2)=\dfrac{10!}{8!}=9\cdot10=90$ ways.