Question

ryan has 5 pairs of pants, 3 shirts, and 3 sweaters. make a tree diagram on back to show the possible outfits that ryan can make.

Explanation:

Step1: Define the categories

Ryan has three categories of clothing: pants (5 pairs), shirts (3), and sweaters (3). The tree diagram will have three levels: pants, shirts, and sweaters.

Step2: Start with pants

For each of the 5 pairs of pants (let's label them as Pant 1, Pant 2, Pant 3, Pant 4, Pant 5), we will branch out to shirts.

Step3: Branch to shirts

For each pair of pants, we have 3 shirts (Shirt 1, Shirt 2, Shirt 3). So from each pant node, draw 3 branches for the shirts.

Step4: Branch to sweaters

For each shirt node (which is already a branch from a pant node), we have 3 sweaters (Sweater 1, Sweater 2, Sweater 3). So from each shirt node, draw 3 branches for the sweaters.

Step5: Calculate total outfits (optional, but to verify)

The total number of possible outfits can be calculated using the multiplication principle: \( 5\times3\times3 = 45 \). This means the tree diagram will have 5 (pants) × 3 (shirts) × 3 (sweaters) = 45 end - nodes, each representing a unique outfit.

To construct the tree diagram:

• First level: 5 nodes labeled Pant 1, Pant 2, Pant 3, Pant 4, Pant 5.
• Second level: From each Pant node, draw 3 nodes labeled Shirt 1, Shirt 2, Shirt 3.
• Third level: From each Shirt node, draw 3 nodes labeled Sweater 1, Sweater 2, Sweater 3.

For example, starting with Pant 1:

• Pant 1 -> Shirt 1 -> Sweater 1
• Pant 1 -> Shirt 1 -> Sweater 2
• Pant 1 -> Shirt 1 -> Sweater 3
• Pant 1 -> Shirt 2 -> Sweater 1
• Pant 1 -> Shirt 2 -> Sweater 2
• Pant 1 -> Shirt 2 -> Sweater 3
• Pant 1 -> Shirt 3 -> Sweater 1
• Pant 1 -> Shirt 3 -> Sweater 2
• Pant 1 -> Shirt 3 -> Sweater 3

And we repeat this process for Pant 2, Pant 3, Pant 4, and Pant 5.

Answer:

The tree diagram is constructed with 5 pant nodes at the first level, 3 shirt nodes branching from each pant node at the second level, and 3 sweater nodes branching from each shirt node at the third level. The total number of possible outfits (end - nodes) is 45. To draw it, follow the step - by - step branching as described above. If we only consider the structure, the tree has three levels with the number of nodes at each level being 5, \( 5\times3 = 15 \), and \( 5\times3\times3=45 \) respectively.