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.