permutations and computations

Question

a catering service offers 12 appetizers, 7 main courses, and 8 desserts. a customer is to select 7 appetizers, 2 main courses, and 2 desserts for a banquet. in how many ways can this be done ?

Answer 1

My guess is the title of this post was supposed to be permutations and combinations. That is because it is combinations that are used to calculate for example the possible number of appetizers selected. Since there are 12 appetizers to choose from and you are to select any 7 of them we say this is the combination of 12 things taken 7 at a time. ₁₂C₇= 12!/7!(12-7)! = 12(11)(10)(9)(8)(7)(6)/ (7)(6)(5)(4)(3)(2) = 792. Combinations mean order does not matter. Note that ₁₂C₇=₁₂C₅. = 12(11)(10(9)(8)/(5)(4)(3)(2) an easier manual calculation.

Onto the number of combinations of main courses: ₇C₂=7(6)/2 = 21. Do you start to see a pattern? Instead of remembering the complicated looking formula, just start like you are going to write 7! in the numerator, but stop after 2 factors are written. Then divide by 2!.Go back and look at ₁₂C₅ to see if that pattern holds.

For the dessert selection: ₈C₂=8(7)/2=28

So there are 792 ways to select appetizers, 21 ways to select main course, and 28 ways to select dessert.

The fundamental counting principle says to multiply those numbers together to determine how many ways to set the banquet table:

792(21)(28)= ??

Left to you.