If there are 4 choices of salad dressing and 2 choices of ice cream, how many possible dinner combinations are there?

Question

You and a friend are at Paul and Elizabeth's Restaurant for a night of fine dining. You can choose among 2 appetizers, 4 main dishes, and 3 desserts.

Appetizer :soup or salad

Main Dish: fish, chicken, beef, vegetarian

Dessert: chocolate cake, apple pie, ice cream

If there are 4 choices of salad dressing and 2 choices of ice cream, how many possible dinner combinations are there?

Answer 1

The best way to address this problem is with a tree chart. Since this is mostly a typing based platform I'll attempt in another manner to make a tree diagram. Imagine below a line connecting the appetizer to each main dish and then to the dessert.

First we have two options

cake

fish pie

ice cream

cake

chicken pie

ice cream

cake

Soup beef pie

ice cream

cake

veg pie

ice cream

We do the same thing with Salad. To obtain the amount of options we have, we look at the last column and count. Above there are 12 combinations. A chart beginning with salad will also have 12 combinations. So there is a total of 24 combinations.

This can also be achieved by multiplication:

2 appetizers * 4 main dishes * 3 desserts = 2*4*3 = 24 combinations.

Now adding salad dressings and ice cream choices. If there are four choices of salad dressing, then we can say there are 4 different salads and therefore 5 total different appetizers. If there are two choices of ice cream, then there are a total of 4 desserts. By multiplication we have:

5 appetizers * 4 main dishes * 4 desserts = 80 combinations.

Create a tree diagram to verify this.

Answer 2

We can break this problem down into cases:

Since we have more options for salads and ice cream we can separate by those.

1. Lets say we order salad and ice cream: we have 4 choices for dressing, 2 choices for ice cream, and 4 choices for main dish so: 4*2*4 = 32 ways.
2. Another case: order salad and not ice cream: we have 4 choices for dressing, 2 choices for dessert (since not ice cream), and 4 choices for main dish so: 4*2*4 = 32 ways
3. Lets say we order soup and ice cream: we have 1 choice for soup, 4 choices for main dish, and 2 choices for ice cream so: 1*4*2 = 8 ways
4. Another case: order soup and not ice cream: we have 1 choice for soup, 4 choices for main dish, and 2 choices for dessert (since not ice cream) so: 1*4*2 = 8 ways

These are all separate cases so we can add them together to get 32+32+8+8 = 80 different possible dinner combinations.