An employer has $500 that she can award in bonuses to her 7 employees. If she wants all of the bonuses to be in $25 increments and not everyone has to receive bonuses, how many different ways are there for her to give out bonuses?

I still don't understand how to solve this problem?
Why would you use combinations for this?

To solve this problem, we can use combinations because we want to find all the possible ways to distribute the bonuses among the employees.

First, let's consider the possible number of bonuses each employee can receive. Since the bonuses should be in $25 increments, dividing $500 by $25 gives us 20. Therefore, each employee can receive between 0 and 20 bonuses.

Now, we can think of the distribution of bonuses as selecting the number of bonuses each employee would receive. To do this, we can use combinations.

The formula for combinations is given by:

C(n, r) = n! / (r! * (n-r)!)

Where n is the total number of options and r is the number of options chosen.

In this case, we have 21 options (0 to 20 bonuses) for each employee and we need to select 7 employees to receive bonuses. So, applying the formula, we get:

C(21, 7) = 21! / (7! * (21-7)!)
= 21! / (7! * 14!)

Now, let's calculate it step by step:

1) Calculate the factorial of 21: 21! = 21 * 20 * 19 * 18 * 17 * 16 * 15 * 14!
2) Calculate the factorial of 7: 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1
3) Calculate the factorial of 14: 14! = 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

Now, substitute the values in the formula:

C(21, 7) = (21 * 20 * 19 * 18 * 17 * 16 * 15 * 14!) / [(7 * 6 * 5 * 4 * 3 * 2 * 1) * (14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)]

Simplifying this expression will give us the total number of different ways the employer can give out bonuses.