Buy 25 pieces of jelly of 3 flavours with at least 5 of 1 flavour. How many ways to buy the flavours? > Terra, Ventus and Aqua are very good friends, almost like siblings. To celebrate Terra’s job promotion, Ventus and Aqua decided to buy Terra’s favorite dessert, fruit-flavored jelly, particularly cherry-flavored jelly. The two decided to buy a total of 25 pieces of jelly of varying flavors: > > * cherry, > * blueberry and > * kiwi. > > > When they arrived at the jelly market, they find that there are only 9 pieces of jelly left for each of the three flavors that they want to buy. > > Since it’s also Terra’s birthday, they would buy at least 5 pieces of cherry-flavored jelly. > > Now, in how many ways can they pick the flavors for the remaining pieces of jelly?

Let $X_i, i = 1,2,3$ be the number of pieces of jelly from cherry, blueberry and kiwi respectively. They have decided that they buy 5 pieces of cherry for the birthday girl. They also decided that they will buy 25 of which 5 is cherry and the rest 20 from the remanining flavors including cherry. There are only 9 available in each flavor for 2 and 3 and only 4 available for 1 ( as she would scoop 5 already).

$x_i \le 9, i = 2,3$ and $x_1\le4$

Let $y_i = 9-x_i, i = 2,3$ and $y_1 = 4-x_1$ with $y_i's\ge0$

The classic problem is $x_1+x_2+x_3 = 20$

or $4-y_1+9-y_2+9-y_3 = 20$

or $y_1+y_2+y_3 = 2$

The number of solutions for this would be the number of ways they could buy the 20 jellies which is ${(2+3-1)\choose(3-1)} = 6$