+127 Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?
+997 Precise solution right here
https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_10A_Problems/Problem_21
This question is from an AMC 10A math competition. I like those...they're fun.
+997 And... if you just want a written out solution...
Suppose the six cookies to be chosen are the stars, as we attempt to implement stars and bars.
We take two dividers, and place them between the cookies, such that the six cookies are split into 3 groups, where the groups are the number of chocolate chip, oatmeal, and peanut butter cookies, and each group can have 0.
First, assume that the two dividers cannot go in between the same two cookies.
By stars and bars, there are \(\binom{7}{2}\) ways to make the groups.
Finally, since the two dividers can be together, we must add those cases where the two dividers are in the same space between cookies. There are 7 spaces, and hence 7 cases.
Our final answer is\(21 + 7 = \boxed{28}\) assortments.
