Pat is to select 6 cookies from a tray containing only chocolate chip, oatmeal
& peanut butter cookies. There are at least 6 of each of these 3 kinds of
cookies on the tray. How many different assortments of 6 cookies can be
selected?

If Pat chooses 0 (no) chocolate chip cookies, there are 7 possible numbers of
oatmeal cookies she can choose, either 0,1,2,3,4,5, or 6. Then the rest of the
6, if any, will be peanut butter cookies.  That accounts for 7 ways if she
chooses 0 chocolate chip cookies.

If Pat chooses 1 chocolate chip cookie, there are 6 possible numbers of
oatmeal cookies she can choose, either 0,1,2,3,4, or 5. Then the rest of the 6,
if any, will be peanut butter cookies.  That accounts for 6 ways if she chooses
1 chocolate chip cookies.
  
If Pat chooses 2 chocolate chip cookies, there are 5 possible numbers of
oatmeal cookies she can choose, either 0,1,2,3, or 4. Then the rest of the 6,
if any, will be peanut butter cookies.  That accounts for 5 ways if she chooses
2 chocolate chip cookies.

If Pat chooses 3 chocolate chip cookies, there are 4 possible numbers of
oatmeal cookies she can choose, either 0,1,2, or 3. Then the rest of the 6, if
any, will be peanut butter cookies.  That accounts for 5 ways if she chooses 3
chocolate chip cookies.

If Pat chooses 4 chocolate chip cookies, there are 3 possible numbers of
oatmeal cookies she can choose, either 0,1, or 2. Then the rest of the 6, if
any, will be peanut butter cookies.  That accounts for 3 ways if she chooses 4
chocolate chip cookies.

If Pat chooses 5 chocolate chip cookies, there are 2 possible numbers of
oatmeal cookies she can choose, either 0 or 1. Then the rest of the 6, if any,
will be peanut butter cookies.  That accounts for 2 ways if she chooses 5
chocolate chip cookies.

If Pat chooses 6 chocolate chip cookies, there is only 1 possible number of
oatmeal cookies she can choose, namely 0. Then the rest of the 6 will be peanut
butter cookies.  That accounts for 1 way if she chooses 6 chocolate chip
cookies.

So the answer is gotten by adding these up

7 + 6 + 5 + 4 + 3 + 2 + 1 = 28 ways.

Here they all are:

 1.  0 chocolate chip, 0 oatmeal and 6 peanut butter cookies.
 2.  0 chocolate chip, 1 oatmeal and 5 peanut butter cookies.
 3.  0 chocolate chip, 2 oatmeal and 4 peanut butter cookies.
 4.  0 chocolate chip, 3 oatmeal and 3 peanut butter cookies.
 5.  0 chocolate chip, 4 oatmeal and 2 peanut butter cookies.
 6.  0 chocolate chip, 5 oatmeal and 1 peanut butter cookies.
 7.  0 chocolate chip, 6 oatmeal and 0 peanut butter cookies.
 8.  1 chocolate chip, 0 oatmeal and 5 peanut butter cookies.
 9.  1 chocolate chip, 1 oatmeal and 4 peanut butter cookies.
10.  1 chocolate chip, 2 oatmeal and 3 peanut butter cookies.
11.  1 chocolate chip, 3 oatmeal and 2 peanut butter cookies.
12.  1 chocolate chip, 4 oatmeal and 1 peanut butter cookies.
13.  1 chocolate chip, 5 oatmeal and 0 peanut butter cookies.
14.  2 chocolate chip, 0 oatmeal and 4 peanut butter cookies.
15.  2 chocolate chip, 1 oatmeal and 3 peanut butter cookies.
16.  2 chocolate chip, 2 oatmeal and 2 peanut butter cookies.
17.  2 chocolate chip, 3 oatmeal and 1 peanut butter cookies.
18.  2 chocolate chip, 4 oatmeal and 0 peanut butter cookies.
19.  3 chocolate chip, 0 oatmeal and 3 peanut butter cookies.
20.  3 chocolate chip, 1 oatmeal and 2 peanut butter cookies.
21.  3 chocolate chip, 2 oatmeal and 1 peanut butter cookies.
22.  3 chocolate chip, 3 oatmeal and 0 peanut butter cookies.
23.  4 chocolate chip, 0 oatmeal and 2 peanut butter cookies.
24.  4 chocolate chip, 1 oatmeal and 1 peanut butter cookies.
25.  4 chocolate chip, 2 oatmeal and 0 peanut butter cookies.
26.  5 chocolate chip, 0 oatmeal and 1 peanut butter cookies.
27.  5 chocolate chip, 1 oatmeal and 0 peanut butter cookies.
28.  6 chocolate chip, 0 oatmeal and 0 peanut butter cookies.
 
Edwin
AnlytcPhil@aol.com