AUTO ANSWER:
Let’s try to count all the possibilities in an organized manner. Alexander can get 12 of one type of cookie in 3 ways. He can get 11 of one type and 1 of another type in 3 ways. If there are 12 glazed, then there is only 1 way to complete the order with chocolate and cherry. If there are 11 glazed, then there are 2 ways (1 chocolate or 0 chocolate). If there are 10 glazed, then there are 3 ways. And so on, until if there are 0 glazed, then there are 13 ways (anywhere from 12 to 0 chocolate). So the answer is 1 + 2 + … + 13 = 13 * 14 / 2 = 91 assortments. Alternative solution: Let’s, instead, use the counting technique known as “stars and bars.” The idea is that we arrange 12 stars to represent the cookies and two bars to separate the cookies into the three different categories. For example, the arrangement ****|**|****** represents the possibility that Alexander buys 4 of the first type of cookie, 2 of the second type, and 6 of the third type. Our question now becomes: How many ways can we arrange the 12 stars and 2 bars? Thus, the number of assortments of a dozen cookies he can buy is 14C2 = 14!/(12! × 2!) = (14 × 13)/(2 × 1) = 7 × 13 = 91 assortments.