In mr tarkiainen's history class each student chooses two of the topics listed in the chart above for their papers. how many different pairs can juliana choose? image: a. 90 b. 10 c. 45 d. 30

There are a total of 10 topics, and we know that each student can choose 2. For example, starting with Steamboat, we can see all of the options that she can choose with the option of Steamboat:

Steamboat, Photography
Steamboat, Telegraph
Steamboat, Sewing Machine
Steamboat, Hydroelectric Power
Steamboat, Telephone
Steamboat, Light Bulb
Steamboat, Rabies Vaccine
Steamboat, Diesel Engine
Steamboat, Airplane

Total = 9

Then, we can list all of our options for the next topic:

Photography, Telegraph
Photography, Sewing Machine
Photography, Hydroelectric Power
Photography, Telephone
Photography, Light Bulb
Photography, Rabies Vaccine
Photography, Diesel Engine
Photography, Airplane

Total = 8

As we see, the total number dropped by one. With this pattern continuing, our totals would look like this:

Steamboat: 9
Photography: 8
Telegraph: 7
Sewing Machine: 6
Hydroelectric Power: 5
Telephone: 4
Light Bulb: 3
Rabies Vaccine: 2
Diesel Engine: 1
Airplane: 0

To find the total number of combinations she has, we add all of our totals together.

9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 45

Juliana can choose 45 different pairs.