To determine the number of ways the three planning duties can be assigned to the employees, we use the permutation rule, since the order in which the duties are assigned matters.

Step-by-Step Solution:

1. Select an employee for the first duty (reserving a venue): There are 10 eligible employees. Thus, there are 10 choices for the first duty.

2. Select an employee for the second duty (arranging catering): After the first employee has been assigned, there are $10 - 1 = 9$ remaining employees. Thus, there are 9 choices for the second duty.

3. Select an employee for the third duty (planning activities): After the first two employees have been assigned, there are $10 - 2 = 8$ remaining employees. Thus, there are 8 choices for the third duty.

4. Calculate the total number of permutations: Multiply the number of choices for each duty: $\text{Total number of ways} = 10 \times 9 \times 8 = 720$

Final Answer:

There are 720 different ways to assign the duties to the employees.

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Would you like me to go over the concept of permutations or provide another example? 😊

Related Questions:

1. What if there were 15 employees instead of 10? How many ways would the duties be assigned?
2. How does this problem differ if the order of duties does not matter?
3. If one specific employee must reserve the venue, how many ways can the remaining duties be assigned?
4. What formula is generally used to calculate permutations?
5. Can this problem be solved using a combination instead of a permutation? Why or why not?

Tip:

When deciding between permutations and combinations, remember that permutations account for order, while combinations do not.