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 don't understand how to approach this problem? Is it by using combinations?
We know that 500 = 25*20 which is the number of partitions of 20 into 7 parts:
20 0 0 0 0 0 0 * 7!/6!
19 1 0 0 0 0 0 * 7!/5!
...
4 4 4 3 3 1 1 * 7!/3!2!2!
...
However, do we have to calculate all of the 20 partitions and then add up the permutations?
To approach this problem, you can use a combination of partitioning and permutations.
First, let's consider the number of partitions of 500 into 7 parts. Each part represents the amount of bonus given to each employee. Since the bonuses are in $25 increments, we need to find the number of ways to distribute 500 as a sum of 7 whole numbers that are divisible by 25.
To simplify this, let's divide everything by 25. Now we need to find the number of ways to distribute 20 as a sum of 7 non-negative integers. This can be thought of as placing 7 identical balls into 20 distinct bins, where each bin represents one of the possible values for a bonus (0, 1, 2, ..., 20).
To count the number of ways, we can use a technique called "stars and bars". Imagine representing each bonus as a star (7 stars in total) and the bins as spaces between bars (19 spaces in total). For example, the distribution "4 4 4 3 3 1 1" can be represented as "****|****|****|***|***|*|*". There is a one-to-one correspondence between such representations and the distributions.
Now, we need to calculate the number of ways to place the 7 stars among the 19 spaces. This can be done using combinations. The formula for combinations is C(n, k) = n! / (k! * (n-k)!), where n is the total number of spaces and k is the number of stars.
In this case, the number of ways to distribute the bonuses can be calculated as C(19, 7), which equals 50388.
Therefore, there are 50388 different ways for the employer to give out the bonuses.
Note: There's no need to calculate all the 20 partitions and then add up the permutations. You can directly calculate the combinations as described above.