A restaurant offers a take-out deal where any combination of 10 items can be purchased for $5. The menu choices are divided into five types: Tacos, Burritos, Tamales, Chiles Rellenos, and Chimichangas. For example, a person could order 2 Tacos, 4 Burritos, a Tamal and 3 Chimichangas. Or maybe 7 Tamales and 3 Chiles Rellenos. Or perhaps 10 Chimichangas and nothing else.
How many different ten-item combinations are available?
Think about 14 items, 4 of which are separators (X's) between types and
the 10 of which are actual items (O's). The items are always
ordered with Tacos first, then Burritos, then Tamales, then Chille
Rellenos, then Chimichangas.
In this scheme, 2 Tacos, 4 Burritos, a Tamale, and 3 Chimichangas would be represented as OOXOOOOXOXOOO
7 tamales and 3 Chile Rellenos would be represented as: XXOOOOOOOXOOOX
10 Chimichangas would correspond to: XXXXOOOOOOOOOO
So this problem is equivalent to determining all the different arrangement of 4 seperators among 14 spaces.
This equals C(14,4) = 14! / 4! * 10 ! =
14*13*12*11/(4*3*2*1) = 1001