How do I make solve this problem with a tree diagram?

Question

Here's the whole problem:

Assume that there are 6 types of upholstery, 6 types of wood, and 3 designs to choose from. Of the designs, 2 designs permit both a choice of wood and a choice of upholstery, but 1 of the design allows only a choice of upholstery.
How many different chairs can be made?

 

I thought to make the tree, you start with the three designs, the first three branches being 1, 2, and 3. Then branching off of the first design is two branches (w) and (up), the same for the second design, then for the third branch just a branch for (up). I'm not sure how to find the answer for this problem.

 

 

 

Answer 1

Just count the number of end-nodes.

 

Design 1...6 choices for wood, 6 choices for upholstery.   This is 36 combinations, altogether.

Design 2...also choices for wood, 6 choices for upholstery. This is 36 combinations, altogether. So far, 72 possibilities.

Design 3...only can choose an upholstery...This is 6 more possibilities.

 

Total: 78 distinct permissible choices of design & refinements.

Answer 2

Hi Alexis,

 

I think you have the right idea but let's see what we can do.

 

Start with the 3 branches for the 3 designs.

I'm thinking of starting from the top of the page and working down.

The first 2 branches have 2 branches each (wood or upholstery)

The 3rd branch has 1 branch, upholstery

 

Then count the final branches.  You should have 5 branches and 5 different chairs that can be made.

Design 1, wood

Design 1, upholstery

Design 2, wood

Design 2, upholstery

Design 3, upholstery