2009 HMMT November Guts Round Problem 13

8 students are practicing for a math contest, and they divide into pairs to take a practice test. In how many ways can they be split up?

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2009 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.