From a committee of 8 people, in how many ways could we elect a 3 person executive board consisting of president, vice president, and treasurer?  Committee members may not hold more than 1 executive role. (Your answer should be a number.)

To solve this, we need to determine the number of ways to assign three different roles (president, vice president, and treasurer) from a group of 8 people. Since the order matters (the roles are distinct), this is a permutation problem.

The formula for the number of permutations is:

\[
P(n, r) = \frac{n!}{(n - r)!}
\]

where:
- \(n\) is the total number of people (8 in this case),
- \(r\) is the number of roles to be filled (3 in this case).

Plugging in the values:

\[
P(8, 3) = \frac{8!}{(8 - 3)!} = \frac{8!}{5!}
\]

Since \(8! = 8 \times 7 \times 6 \times 5!\), the \(5!\) cancels out, leaving:

\[
8 \times 7 \times 6 = 336
\]

Thus, there are **336** ways to elect a 3-person executive board from a committee of 8 people.

Would you like more details on this, or do you have any questions?

Here are 5 related questions to consider:
1. How would the number of ways change if the committee had 10 people instead of 8?
2. What if we were choosing a 3-person subcommittee, where the order doesn't matter?
3. How many ways are there to choose 4 people for roles from a committee of 10?
4. What is the difference between combinations and permutations in problems like these?
5. How can the same concept be applied to choosing teams in a sport?

**Tip:** In permutation problems, the order of selection matters, while in combination problems, it does not.

From a committee of 8 people, in how many ways could we elect a 3 person executive board consisting of president, vice president, and treasurer? Committee members may not hold more than 1 executive role.

Learn how to calculate the number of ways to elect a 3-person executive board (president, vice president, and treasurer) from a committee of 8 using the permutation formula P(n, r). This detailed guide explores the mathematical concepts of permutations and counting principles, suitable for students in Grades 9-12.

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