Sure! Let's solve the problem step-by-step.
We need to find the probability that you and your friend are both chosen out of a group of 6 people when 3 contestants are selected.
1. Total Number of Ways to Choose Contestants:
The total number of ways to choose 3 contestants out of 6 is given by the combination formula:
[tex]\[
\binom{6}{3} = 20
\][/tex]
This means there are 20 different ways to choose 3 people out of 6.
2. Choosing You and Your Friend:
If you and your friend are to be chosen, that already uses up 2 out of the 3 spots. We need to choose 1 more contestant from the remaining 4 people.
3. Number of Ways to Choose the Remaining Contestant:
The number of ways to choose 1 additional contestant from the remaining 4 people is given by another combination:
[tex]\[
\binom{4}{1} = 4
\][/tex]
This means there are 4 ways to choose 1 remaining contestant from the 4 people left.
4. Calculating the Probability:
The probability is the ratio of the number of favorable outcomes (choosing you and your friend and one more person) to the total number of possible outcomes (choosing any 3 people from 6).
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{4}{20}
\][/tex]
5. Simplifying the Fraction:
Simplifying [tex]\(\frac{4}{20}\)[/tex]:
[tex]\[
\frac{4}{20} = \frac{1}{5}
\][/tex]
However, the probability in fractions before simplification is preferred to match the options given:
So, the probability that you and your friend are both chosen is:
[tex]\[
\frac{4}{20}
\][/tex]
Therefore, the correct answer is:
B. [tex]\(\frac{4}{20}\)[/tex]
