Famous Arithmetic property of lotteries - restricted partition of integer into exactly k distinct parts between a given set

Question

I would like to find a complete explanation regarding a famous arithmetic property of lotteries :

Let´s say a friend of us is a regular player of a lottery where 5 numbers are taken from a box containing 50 numbers from 1,2,3....50. When a number is drawn, it is not replaced into the box which implies that there is no reptition allowed.

The property I want to discuss is the following: the sum of the five drawn numbers is not equiprobable. Indeed, there is just one combination that would give 15 when summed up: 1,2,3,4,5. Whereas there are a lot more that would give 120 for instance. Hence, people not familiar with combinatory logic are easily biased and lured to choose numbers according to the sum they would induced.

It boils down to a partition problem, with restrictions. There is a lot of litterature since Euler on this topic, but I have not find the perfect solution yet. The generating functions are easy to find for partitions into up to k parts, or into k distinct parts and so on. None of them have all the restrictions I want.

Thus, the final question is:
- what is the number of partitions of an integer n into exactly K distinct parts taken into [1;q] ? In order to be specific, let's say: how many combinations of numbers from the lottery described above would give 120 ? I precise that it is easy to compute the solution with a little program but it is not what we are looking for.

Thanks all,

Golani.