In-Class Activity: Introduction to Counting

Consider the All Day Menu at Noonie’s Deli here in Middlebury, VT.

This menu includes three main sections:

Our Favorite Sandwiches, a list of pre-determined sandwiches that you can order as described.
Our Favorite Salads, a list of pre-determined salads that you can order as described.
Make Your Own, in which you can assemble a custom sandwich by choosing your main ingredient, bread/wrap, cheese, vegetables, and dressing.
- For the purposes of this problem, ignore the “Choose Any Extras,” “Drinks,” and “Baked Goods” sections.

For the purposes of this problem, a meal means either a sandwich or salad.

You can ignore the advertised possibility of 1/2 sandwiches.

Use the menu and the principles of multiplication, addition/summation, and inclusion-exclusion to answer the following questions.

Part A

Nia and Marcus are going to order a meal from Noonie’s. Nia only eats sandwiches, but doesn’t want to go through the hassle of designing her own. Marcus only eats salads.

How many distinct ways are there for Nia and Marcus to order their meal?

Part B

Luis likes both salads and sandwiches, but doesn’t want to go through the hassle of designing their own. He’ll order either a salad or a sandwich, but not both.

How many ways are there for Luis to order a meal?

Part C

Maia wants to make her own custom sandwich. She’s happy with any of the main ingredients, but doesn’t want to pay extra for premium breads, additional vegetables, or extras.

How many ways are there for Maia to order her custom sandwich?

Part D

Linh is vegan and doesn’t want to pay for any extras.

How many ways are there for Linh to order her custom sandwich?

Part E

Hiroki wants to make a custom sandwich. He’s willing to have one with either a premium bread OR an extra (like avocado or hummus), but not both. He’s also willing to make a custom sandwich with neither a premium bread nor extras. Hiroki doesn’t want any extra vegetables.

How many ways are there for Hiroki to order his custom sandwich?

Part F

Aryana wants a custom sandwich without any premium breads, extras. She thinks that pickled red onions really make roasted tomatoes taste great, and she isn’t willing to order a sandwich that includes only one of these two things: either the sandwich has both pickled red onions and roasted tomatoes, or it has neither.

How many ways are there for Aryana to order her custom sandwich?