The Doughnut Problem
There is a doughnut store that offers six kinds of doughnuts: V anilla, C hocolate, B lueberry, S trawberry, L emon, and P each. The Super Special is a dozen doughnuts of your choice, mixed or matched however you like. The order you chose the doughnuts doesn't matter.
How many different Super Specials are there?
(Note: this problem was taken from Life of Fred: Advanced Algebra, then rewritten.)
The answer is 6188.
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1 solution
Imagine you get home and lay all the donuts in your Super Special out on a table. You group them together by kind and then, after each row, you put a black rubber donut. These black rubber donuts act like bookmarks between the kinds of donuts.
So, if you had 12 Lemons, it would look like this:
R R R R LLLLLLLLLLLL R
In every example, you will end up with 17 slots (12 donuts + 5 rubber donuts.)
Now we take out the edible donuts and just leave the R s. So my example looks something like this:
R R R R _ _ _ _ _ _ _ _ _ _ _ _ R
Every Super Special has a unique combination of R s and slots. So now, we just figure out how many ways you can pick five of the 17 slots (to put the R s in):
1 2 ! 5 ! 1 7 ! = 6188