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?
500/25 = 20 ... +1 = 21
C(21,7) = 116,280
16 letters .. C, M, O, and E repeat
12!*2^4 = 7664025600