Cookie Problem!

There are a total of 4+4+2=10 cookies. We want to pick 6 of them, so the total number of ways to do this is (610​)=10!/6!4!​=210.

However, we need to divide this number by 3! because the order in which we pick the cookies does not matter. (For example, picking a chocolate chip cookie, then an oatmeal cookie, and then a peanut butter cookie is the same as picking a peanut butter cookie, then an oatmeal cookie, and then a chocolate chip cookie.) So the final answer is 210/3!=35​.

Partition of 10 cookies into 6 as follows:

 

1 = (4, 2, 0)
2 = (3, 3, 0)
3 = (2, 4, 0)
4 = (4, 1, 1)
5 = (3, 2, 1)
6 = (2, 3, 1)
7 = (1, 4, 1)
8 = (4, 0, 2)
9 = (3, 1, 2)
10 = (2, 2, 2)
11 = (1, 3, 2)
12 = (0, 4, 2)
Total number of ways== 12 ways to pick 6 cookies out of 10