The question asks how many ways four positions (president, vice president, secretary, and treasurer) can be filled from a group of nine people.

Since these are distinct positions, order matters, so this is a permutation problem. The number of ways to choose and arrange 4 positions from 9 people is calculated using the permutation formula:

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

Here, $n = 9$ (the total number of people) and $r = 4$ (the number of positions to fill).

$P(9, 4) = \frac{9!}{(9 - 4)!} = \frac{9!}{5!}$

This simplifies to:

$P(9, 4) = 9 \times 8 \times 7 \times 6 = 3024$

So, there are 3024 ways to fill the four positions.

Would you like further clarification or more details on this? Also, here are five related questions to think about:

1. What if there were only three positions instead of four?
2. How would the answer change if two positions were identical (e.g., two vice presidents)?
3. What if there were 10 members instead of 9—how many ways could the positions be filled?
4. Can you think of a scenario where combinations, instead of permutations, would be more appropriate?
5. How does the answer change if some positions have restrictions (e.g., only certain people can be president)?

Tip: In permutation problems, remember that order matters. For combinations, order does not matter.