There are $6$ types of cookies. How many different packs of $3$ cookies can the baker package? > A cookie baker packages cookies $3$ to a pack. The type of cookies she can choose from include chocolate chip, oatmeal, sugar-coated, sugar-free, peanut butter, and hazelnut. How many different packs of $3$ cookies can she package? When I approached the problem, I assumed that you had as many of each cookies as you wanted. So the first way to choose the first cookie is $6$, the second is $6$, and the third is $6$ as well. So the total number of possible permutations is $216$. To resolve the order, divide by $3!$ which then yields a total of $36$. The answer is however $56$ possible packages. I would appreciate that you not only give you provide the correct analysis and solution, but also where I made a mistake.

We assume the order of cookies in the package doesn't matter.

There are $\binom{6}{3}$ three-flavour packages. To count the two-flavour packages, the majority cookie can be chosen in $6$ ways, and for each of these ways the minority cookie can be chosen in $5$ ways. Finally, there are $6$ one-flavour packages, for a total of $56$.

_Remark:_ Dividing $216$ by $3!$ is not right. The division by $3!$ is correct for three-flavour packages, but two-flavour ordered packages only come in $3$ orders, not $3!$, and one-flavour packages come in only one order.