Dividing flowers between two girls The problem statement is:
Two girls have picked 10 Roses  , 15 Sunflowers  and 14 Daffodils  . Find the number of ways they can divide the flowers between themselves.
My approach:
Let there is a single stick that divide the flowers. Total number of objects including the stick is 40, among which 10 objects(roses) are of one kind, 15 objects(sunflowers) of another kind and 14 objects (daffodils) are of yet another kind. Hence total number of possible distributions should be 40!/(10! * 15! * 14!). But this is a huge number. The actual answer is 11 * 16 * 15.
What is wrong with my approach to this problem?
 A: In your method, you have calculated the number of unique permutations of the flowers. This method can be used to find the number of distinct permutations of the word MISSISSIPPI (Wikipedia), but this is not what you are supposed to calculate.
The problem is that the order of the flowers does not matter here. If one girl has for example 3 roses and 2 sunflowers, one possible combination could be 1 sunflower, 3 roses, and then another sunflower, which is the same as all possible combinations of 3 roses and 2 sunflowers.
A: I don't understand your method at all.
The only thing that matters is how many flowers of each kind the first girl gets. There are $11$ possible options for the number of Roses she gets (the possible numbers are $0,1,2,...,10)$, $16$ possible options for the number of Sunflowers, $15$ options for the number of Daffodils. By the product rule there are $11\times 16\times 15$ options.
A: Let's say you have this permutation of roses (R), sunflowers (S), daffodils (D): RDR|SR, with '|' being the stick (I decreased the total number of flowers, but it's the same idea).
Then, your counting also counts RDR|RS as a different distribution, which is wrong. We don't care about the order of flowers one has, only the amounts of each flower.
The right way to count is, first pick the amount of roses that the first girl will take (there are $11$ options), then the amount of sunflowers, and finally the amount of daffodils.
