This is the same as the shake-hand problem.
Consider when two students pair-up to do a project, they are shaking hands, i.e. the handshakes do not get counted twice, one from each student.
So with 18 students, each students gets to pair up with 17 other students to make 18*17=306 projects.
However, since we don't count each project twice, we end up with 306/2=153 projects.
