A high school has 10 students who want to be class officers. If there are 4 different positions open, how many ways are there for the students to fill them?

This is a permutation problem because the order in which the students are selected matters (i.e., being chosen as president is different from being chosen as vice president, etc.).

The formula for permutations is:

P(n, r) = n! / (n-r)!

where:

• n is the total number of options,
• r is the number of options chosen at a time, and
• "!" denotes a factorial, meaning the product of all positive integers up to that number.

In this case, n = 10 (the number of students) and r = 4 (the number of positions).

So, the calculation would be:

P(10, 4) = 10! / (10-4)!

= 10987654321 / 654321

= 1098*7

= 5040

So, there are 5040 different ways for the students to fill the 4 positions.