Question

a landscaper is selecting two trees to plant. he has five to choose from. three of the five are deciduous and two are evergreen. what is the probability that he chooses trees of two different types? express your answer as a percent. 30% 40% 50% 60%

Explanation:

Step1: Calculate total number of ways to choose 2 trees from 5.

The formula for combinations is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 5 \) and \( k = 2 \).
\( C(5, 2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4\times3!}{2\times1\times3!}=10 \)

Step2: Calculate number of ways to choose 1 deciduous and 1 evergreen tree.

Number of ways to choose 1 deciduous from 3: \( C(3, 1)=\frac{3!}{1!(3 - 1)!}=3 \)
Number of ways to choose 1 evergreen from 2: \( C(2, 1)=\frac{2!}{1!(2 - 1)!}=2 \)
By the multiplication principle, number of ways to choose 1 deciduous and 1 evergreen: \( 3\times2 = 6 \)

Step3: Calculate the probability.

Probability \( P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{6}{10}=0.6 \)
To convert to a percent, multiply by 100: \( 0.6\times100 = 60\% \)

Answer:

60%