Mock AIME 1 2007-2008 Problems/Problem 10
Problem 10
An oreo shop sells 
different flavors of oreos and 
different flavors of milk. Alpha and Beta decide to purhcase some oreos. Since Alpha is picky, he will not order more than 1 of the same flavor. To be just as weird, Beta will only order oreos, but she will be willing to have repeats of flavors. How many ways could they have left the store with 3 products collectively? (A possible purchase is Alpha purchases 1 box of uh-oh oreos and 1 gallon of whole milk while Beta purchases 1 bag of strawberry milkshake oreos).
Solution
We use casework:
- Alpha purchases
items, Beta 
. Then there are 
ways for Alpha to pick different items. - Alpha purchases
items, Beta 
. Then there are 
ways for Alpha to pick different items and there are 
ways for Beta to choose her oreo, with a total of 
. - Alpha purchases items, Beta . Then there are
ways for Alpha to pick different items. There are 
ways for Beta to choose two distinct oreos, and ways for Beta to choose two oreos of the same flavor. This totals to 
. - Alpha purchases items, Beta . There are
ways for Beta to choose three distinct oreos. For Beta to choose two oreos of the same flavor and another oreo, there are choices for the first and 
choices for the second, with a total of 
choices. There are ways to choose three of the same flavored oreo. This totals to 
.
. Then there are 
ways for Alpha to pick 
items, Beta 
. Then there are 
ways for Alpha to pick 
.
ways for Alpha to pick 
ways for Beta to choose two distinct oreos, and 
.
ways for Beta to choose three distinct oreos. For Beta to choose two oreos of the same flavor and another oreo, there are 
choices for the second, with a total of 
choices. There are 
.The total is 
.
See also
| Mock AIME 1 2007-2008 (Problems, Source) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
