How many different student body governments are possible if there are 8 seniors, 4 juniors, and 3 sophomores running for the student body offices of senior class president, junior class president, and sophomore class president?

Step 1 ：The problem is asking for the number of ways to choose a senior, a junior, and a sophomore from their respective groups. This is a permutation problem because the order in which we choose the students matters (i.e., choosing a senior first and then a junior is different from choosing a junior first and then a senior).

Step 2 ：The number of ways to choose a senior is the number of seniors, which is 8. Similarly, the number of ways to choose a junior is the number of juniors, which is 4, and the number of ways to choose a sophomore is the number of sophomores, which is 3.

Step 3 ：To find the total number of ways to choose a student body government, we multiply the number of ways to choose a senior, a junior, and a sophomore. So, the total number of ways is \(8 \times 4 \times 3 = 96\).

Step 4 ：Final Answer: The total number of different student body governments possible is \(\boxed{96}\).