So i have 6 different flavours.
I intend to mix these flavours together where the total amount would be 100 % when all the 6 different flavours has been counted.
This means for example that one combination can look like this flavour 1 can have 20 % in there, flavour 2 = 40 %, flavour 3 = 15 %, flavour 4 = 15 %, flavour 5 = 6 %, flavour 6 = 4 %. That setup would together make 100 % and 1 combination has been made.
How do i calculate how many possible combinations you can make with this? I am only interested in how to calculate "full numbers" so no 4,5 % for instance.
Best Answer
The answer depends on whether you require each of the six flavours to be present (that is, at least 1%).
Assuming that you do indeed intend each flavour to be present: Consider 100 little cups waiting to take their flavour portion, representing 1% each. Now insert marks between the cups to show where to change to the next flavour - you need 5 such marks, to insert into 99 different positions.
Thus the answer is $\binom {99}{5} = \frac{99!}{94!\cdot5!}$.
For the other case where some flavours can be absent, this means you can have multiple marks in one space. Now the result looks like arrange 100 cups and 5 marks, which means we have 105 places and we choose 5 of those places to be flavour-change marks.
For this case the answer is $\binom {105}{5} = \frac{105!}{100!\cdot5!}$.