I have the following problem: "There are 15 identical candies. How many ways are there to distribute them among 7 kids?"
I know the conventional way to solve a similar problem (unordered with repetition) is using the formula (n+k−1Ck) (=54,264) and I'm convinced with why this formula works. However, I'm trying to use another approach and I can't figure out why that doesn't work.
Please consider this other problem first: "There are 4 people and 9 different assignments. We need to distribute all assignments among people. No assignment should be assigned to two people. Every person can be given arbitrary number of assignments from 0 to 9. How many ways are there to do it?"
This is a 'Tuples' problem ie. ordered with repetitions and its solution is simply 4^9. Therefore, why can't I solve the first problem with a similar approach, but then dividing by n!
This is how I'm thinking: We have 15 candies and 7 kids. The first candy can be distributed in 7 ways (one of the 7 kids in going to take it). The second can also be distributed in 7 ways .. etc
So we now have 7^15 just like the approach we used for the second (assignments) problem. But now since the candies are identical, order doesn't matter, and we should divide by 15! .. But solving it this way gives a completely different and significantly smaller answer 4,747,561,509,943 / 1,307,674,368,000 = 3.63
What is wrong with thinking about the problem in this approach and why is it not working? I would just like to know what I'm missing here and learn from my mistake so that I don't approach a future problem with a similar thinking.