We have a combinatorics problem. Four students need to sign up for three different sports projects. Each student can only choose one project. We need to find the total number of different ways they can sign up.
Let's analyze this step by step. Each student independently chooses one project from three available options. Student 1 has 3 choices, Student 2 has 3 choices, Student 3 has 3 choices, and Student 4 also has 3 choices. The key insight is that each student's choice is independent of the others.
Using the multiplication principle, we multiply the number of choices for each student. Since each student has 3 independent choices, the total number of ways is 3 times 3 times 3 times 3, which equals 3 to the power of 4, which equals 81.
Let's verify our method with a simpler example. If we have 2 students and 2 projects, we expect 2 squared equals 4 ways. We can list all combinations: Student 1 chooses Basketball and Student 2 chooses Basketball, Student 1 chooses Basketball and Student 2 chooses Soccer, Student 1 chooses Soccer and Student 2 chooses Basketball, and Student 1 chooses Soccer and Student 2 chooses Soccer. That's exactly 4 combinations, confirming our multiplication principle works correctly.
In conclusion, for the original problem of 4 students signing up for 3 sports projects, where each student can only choose 1 project, we use the multiplication principle. Each student has 3 independent choices, so the total number of different ways is 3 to the power of 4, which equals 81. Therefore, there are 81 different ways for the students to sign up.