SOLUTION: A Mix-n-Match selection of 10 cookies is to be made from three varieties: oatmeal, peanut butter, and sugar. How many different selections are possible?

Make a row of 10 stars "*" and 2 bars "|".  Here is one way this could be:

***|*****|** 

The number of stars to the left of the leftmost bar is the number of oatmeal
cookies.
The number of stars between the leftmost bar and the rightmost bar is the number
of peanut butter cookies.
The number of stars to the right of the rightmost bar is the number of sugar
cookies.

***|*****|**  represents 3 oatmeals, 5 peanut butters, and 2 sugars.

***||*******  represents 3 oatmeals, 0 peanut butters, and 7 sugars.

**********||  represents 10 oatmeals, 0 peanut butters, and 0 sugars.

There are 12 things counting both stars and bars.  The 10 stars are
indistinguishable and the 2 bars are also indistinguishable. So there are

 = 66 distinguishable rows of 10 stars and 2 bars, and
therefore 66 possible cookie selections.

Edwin

It is the same as to ask


    +-------------------------------------------------------------+
    |  how many solutions in integer non-negative numbers         |
    |                                                             |
    |  this equation does have   +  +  = 10.            |
    +-------------------------------------------------------------+


The "stars and bars" method gives the answer: the number of solutions for this equation is

     =  =  =  =  = 6*11 = 66.    ANSWER


In this formula, n= 10, k = 3.