Task number: 3865
In the following problems, first express the solution value as a coefficient for some power of \( x \) in a suitable product of polynomials.
Then try to calculate the appropriate coefficient.
The pastry shop sells three types of desserts – pinwheels, canapes and punch cakes. In how many ways can you buy 12 desserts so that at least two desserts are bought of each type but at most three canapes?
How many ways can 10 identical balloons be distributed between two boys and two girls so that each boy receives at least one balloon and each girl at least two?
The beekeeper asks the orchardist to plant 25 new trees, which he has only 4 species available. Orchardist’s wife refuses a walnut tree because it is large and occupies a lot of space. (In addition, everybody knows that walnut honey can’t be eaten :)) Apple trees are also against her taste, they already have too many. On the other hand, she loves infinitely cherry-plum jam, so she imposes harsh conditions – at most one walnut tree, not more than 10 apple trees, at least 6 cherry trees and at least 8 plum trees (slivovitz – an intensive motivation) or divorce. In how many ways can the orchardist salvage his matrimony? (I.e. in how many different ways he can choose the tree species?)