How many different menus showing 10 main courses can a restaurant make if it has 15 main courses from which to choose?

To determine the number of different menus showing 10 main courses from a selection of 15, we use the combination formula ( C(n, r) = \frac{n!}{r!(n-r)!} ), where ( n ) is the total number of options and ( r ) is the number of choices. In this case, ( n = 15 ) and ( r = 10 ). Therefore, the number of different menus is ( C(15, 10) = C(15, 5) = \frac{15!}{5! \cdot 10!} = 3003 ). Thus, the restaurant can create 3,003 different menus.