Six students are contesting the election for the presidentship of the students' union. In how many ways can their names be listed on the ballot papers?

Answer

To solve the problem of how many ways the names of 6 students can be listed on the ballot papers, we can use the concept of permutations.

### Step-by-step Solution:

1. **Understanding the Problem**: We need to list the names of 6 students. The order in which the names are listed matters since it is a ballot paper.

2. **Identifying the Formula**: The number of ways to arrange 'n' distinct objects is given by the formula for permutations, which is \( n! \) (n factorial).

3. **Calculating Factorial**: For our problem, we have 6 students. Therefore, we need to calculate \( 6! \).

4. **Calculating \( 6! \)**:
\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]
- First, calculate \( 6 \times 5 = 30 \)
- Next, \( 30 \times 4 = 120 \)
- Then, \( 120 \times 3 = 360 \)
- After that, \( 360 \times 2 = 720 \)
- Finally, \( 720 \times 1 = 720 \)

5. **Final Answer**: Thus, the total number of ways the names of the 6 students can be listed on the ballot papers is \( 720 \).