If you have 4 appetizers and you need to choose just 3 of them, it would be the same amount of options as just choosing 1 out of the 4; so there are 4 ways to choose the appetizer. For the main course, you are choosing 2 out of 6. So there are 6 ways to choose the first main course, and assuming there is no repeats, there would be 5 ways to choose the next one. However, there will be some overlap, so you need to divide it by 2. So you have [tex] \frac{6*5}{2} =15[/tex] options for main course. Finally for dessert, you have 12 desserts to choose from and 5 desserts to choose. So you do a similar thing to the main courses. [tex] \frac{12*11*10*9*8}{5*4*3*2*1} [/tex] would be the equation you would use. Because there are 12 ways to pick the first dessert, 11 ways for the 2nd and so on until you reach 5 desserts. You divide it by 5*4*3*2*1 or 5! (Factorial) because there are overlaps for each of the 5 so you have to account for those. After solving the equation, there are 792 ways to pick desserts So you now know that you have 4 ways of picking the appetizer, 15 ways for the main course and 972 ways to pick desserts. So for options of the full meal, you need to multiply all these values together: [tex]4*15*792=47520[/tex]. Therefore, there are 47,520 possible ways this could be done.
