Answer :
To determine the probability that the landscaper selects two trees of different types, we need to perform a series of steps using combinations and probability calculations. Here’s the detailed, step-by-step solution:
### Step 1: Determine the Total Number of Trees
There are 5 trees in total: 3 deciduous trees and 2 evergreen trees.
### Step 2: Determine the Number of Ways to Select 2 Trees out of 5
We use combinations to find the number of ways to select 2 trees out of 5. The combination formula is given by:
[tex]\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \][/tex]
where [tex]\( \binom{n}{k} \)[/tex] is the number of ways to choose [tex]\( k \)[/tex] items from [tex]\( n \)[/tex] items.
For our case:
[tex]\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{20}{2} = 10 \][/tex]
So, the total number of ways to choose 2 trees out of 5 is 10.
### Step 3: Determine the Number of Ways to Choose 1 Deciduous and 1 Evergreen Tree
To have one of each type, we choose 1 deciduous tree from 3 and 1 evergreen tree from 2. Again, we use combinations:
Ways to choose 1 deciduous tree out of 3:
[tex]\[ \binom{3}{1} = \frac{3!}{1!(3-1)!} = \frac{3 \times 2!}{1 \times 2!} = \frac{3}{1} = 3 \][/tex]
Ways to choose 1 evergreen tree out of 2:
[tex]\[ \binom{2}{1} = \frac{2!}{1!(2-1)!} = \frac{2}{1} = 2 \][/tex]
To find the total number of ways to choose one of each type, we multiply these two results:
[tex]\[ 3 \times 2 = 6 \][/tex]
So, there are 6 ways to choose one deciduous and one evergreen tree.
### Step 4: Calculate the Probability
The probability of choosing one deciduous and one evergreen tree is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{6}{10} = 0.6 \][/tex]
### Step 5: Express the Probability as a Percentage
To convert the probability to a percentage, we multiply by 100:
[tex]\[ 0.6 \times 100 = 60\% \][/tex]
Therefore, the probability that the landscaper chooses trees of two different types is [tex]\( 60\% \)[/tex].
So the correct answer is:
[tex]\[ \boxed{60\%} \][/tex]
### Step 1: Determine the Total Number of Trees
There are 5 trees in total: 3 deciduous trees and 2 evergreen trees.
### Step 2: Determine the Number of Ways to Select 2 Trees out of 5
We use combinations to find the number of ways to select 2 trees out of 5. The combination formula is given by:
[tex]\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \][/tex]
where [tex]\( \binom{n}{k} \)[/tex] is the number of ways to choose [tex]\( k \)[/tex] items from [tex]\( n \)[/tex] items.
For our case:
[tex]\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{20}{2} = 10 \][/tex]
So, the total number of ways to choose 2 trees out of 5 is 10.
### Step 3: Determine the Number of Ways to Choose 1 Deciduous and 1 Evergreen Tree
To have one of each type, we choose 1 deciduous tree from 3 and 1 evergreen tree from 2. Again, we use combinations:
Ways to choose 1 deciduous tree out of 3:
[tex]\[ \binom{3}{1} = \frac{3!}{1!(3-1)!} = \frac{3 \times 2!}{1 \times 2!} = \frac{3}{1} = 3 \][/tex]
Ways to choose 1 evergreen tree out of 2:
[tex]\[ \binom{2}{1} = \frac{2!}{1!(2-1)!} = \frac{2}{1} = 2 \][/tex]
To find the total number of ways to choose one of each type, we multiply these two results:
[tex]\[ 3 \times 2 = 6 \][/tex]
So, there are 6 ways to choose one deciduous and one evergreen tree.
### Step 4: Calculate the Probability
The probability of choosing one deciduous and one evergreen tree is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{6}{10} = 0.6 \][/tex]
### Step 5: Express the Probability as a Percentage
To convert the probability to a percentage, we multiply by 100:
[tex]\[ 0.6 \times 100 = 60\% \][/tex]
Therefore, the probability that the landscaper chooses trees of two different types is [tex]\( 60\% \)[/tex].
So the correct answer is:
[tex]\[ \boxed{60\%} \][/tex]
