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
