Suppose ten students in a class are to be grouped into teams. if each team has two students, how many ways are there to form teams? (the ordering of students within teams does not matter, and the ordering of the teams does not matter.)

There are two solutions for this problem. It depends if order matters or not. As asked in the problem, we have to show both solutions.

The first solution is for the condition that order does not matter. This is a combination problem:

nCr = n!/r! (n-r) !

where n=10 because there are 10 total students, and r=2 because you choose 2 students at a time

nCr = 10!/2! (10-2) ! = 45 ways

The second solution is for the condition that order matters. This is a permutation problem.

nPr = n! / (n-r) ! = 10! / (10-2) ! = 90 ways