Number theoretic calculation of probability problem involving partitions.

This question asks for a specific miscalculation of how large chance that I get at least one call every day of the week if I in total get 12 calls during a week.

If we instead consider the question how to calculate it...

For some reason I think there should exist a number theoretic solution to this. We have the concept of partitions which count the number of ways to sum something up. For example 5 can be written as $$5=4+1$$ or $$5=3+2$$, $$5=3+1+1$$, in total there are 7 such ways to sum, called partitions.

If we place 7 calls over 12 days then we have 5 calls left to place. How can we place these remaining 5 calls over 7 days? Placing 5 calls over 5 days, then partition could help us, but not here. It feels like we should be able to use partitions or some generalization of them to express this.

• What's the probability distribution? In other words, how are people deciding to call you? Can you rephrase the problem in terms of (integer) compositions? – darij grinberg Sep 30 '18 at 19:30
• @darijgrinberg I think it is uniform distribution. Ah yes you are right maybe it is compositions and not partitions. I am quite new on these things. – mathreadler Sep 30 '18 at 19:53