Waddles doughnuts
There are six kinds of doughnuts offered at Waddle's Doughnuts. (If you need names for them, they are Apple, Blueberry, Chocolate, Dusted, Eggplant, and Frosted.) The Waddle's Special is a dozen doughnuts of your choice, mixed or matched any way you like. How many different Waddle's Specials are there?
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Imagine when you get home with your Waddle's Special bag, that you open it up and line up the doughnuts to see how many of each kind you have selected. first line up the Apple doughnuts. There might be any number of A's from zero to twelve. Then after the A's put a black rubber doughnut. then put the B's in the line and put another black rubber doughnut. Then the C's and another black rubber doughnut. continued until we finally end with the F's. so we now have five black rubber doughnuts--sort of like bookmarks--between the various kinds of doughnuts. if I had 2A's, 5B's, 0C's, 1D, 1E and three F's it would look like AArBBBBBrrDrErFFF where r is a black rubber doughnut.12 E's would look like rrrrEEEEEEEEEEEEr. so we have 17 slots with 5 filled with r's and the rest with doughnuts That means that we then have C(17,5) and we're done.
C(17,5) = 17!/12!5! = 6188