Question 1067721: 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?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Since no one else is answering, I will give this one a try.
I you think my reasoning is faulty, and/or my answer is wrong, let me know.
Let  ,  , and  be non-negative integers,
representing the numbers of each type of cookie in a 6-cookie assortment.
with  and of course  .
The possible (x,y,z) triples are
(6,0,0) ,
(5,1,0) ,
(4,2,0) ,
(4,1,1) ,
(3,3,0) ,
(3,2,1) , and
(2,2,2) .
Of course, there is only  possibility for a (2,2,2) assortment,
but other triples represent more than one assortment.
Triples with two different numbers, like (6,0,0), (4,1,1) , and (3,3,0) have  possibilities,
because we have choices for the type of cookie associated with the unrepeated number.
A triple with 3 different numbers, like (5,1,0) , (4,2,0) , and (3,2,1) represents  different possible assortments,
because there are possible permutation of flavors triples.
So, the total number of assortments is
 .
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