Let's solve this step by step.

Step 1: List all possible outcomes

Nelson can pick one of three trees for each of three locations. Therefore, the total number of possible outcomes is: $3^3 = 27$

Step 2: Calculate the probability of planting one oak tree, one cedar tree, and one elm tree

We need to count the number of favorable outcomes where Nelson plants exactly one oak tree (O), one cedar tree (C), and one elm tree (E).

The favorable outcomes are:

OCE, OEC, COE, CEO, EOC, ECO

There are 6 favorable outcomes.

The probability is: $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{6}{27} = \frac{2}{9}$

Step 3: Calculate the probability of planting two oak trees and one elm tree

We need to count the number of favorable outcomes where Nelson plants exactly two oak trees (O) and one elm tree (E).

The favorable outcomes are:

OOE, OEO, EOO

There are 3 favorable outcomes.

The probability is: $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{27} = \frac{1}{9}$

Final Answers

• The probability of planting one oak tree, one cedar tree, and one elm tree is $\frac{2}{9}$.
• The probability of planting two oak trees and one elm tree is $\frac{1}{9}$.

If you have any questions or need further clarification, please let me know!

Here are 5 questions you might ask next:

1. Can you explain more about how to determine the number of favorable outcomes?
2. How do we calculate the total number of outcomes in more complex scenarios?
3. Can you provide examples of other similar probability problems?
4. What are the common types of probability problems involving combinations?
5. How can these probability concepts be applied in real-life situations?

Tip: Always break down the problem into smaller parts and count the number of favorable outcomes carefully. This method will help ensure accuracy in probability calculations.