Jason, Erik, and Jamie are friends in art class. The teacher randomly chooses 2 of the 21 students in the class to work together on a project. What is the probability that two of these three friends will be chosen?
A. 1/105 B. 1/70 C. 4/50 D. 34/140
Number of possible pairs chosen
= C(21,2) = 210
number of ways to choose 2 of the 3 friends = C(3,2) = 3
prob(of stated event) = 3/210 = 1/70
To find the probability that two of these three friends will be chosen, we need to calculate the total number of possible pairs of students and the number of pairs that consist of two of the three friends.
The total number of possible pairs of students can be calculated using combinations. We need to select 2 students out of the 21 students, which can be done using the combination formula:
C(n, r) = n! / (r!(n - r)!)
Here, n represents the total number of students (21) and r represents the number of students we want to select (2).
Using this formula, we can calculate the total number of possible pairs as:
C(21, 2) = 21! / (2!(21 - 2)!) = 21! / (2!19!) = (21 * 20) / (2 * 1) = 210
Now, let's calculate the number of pairs that consist of two of the three friends: Jason, Erik, and Jamie. Since each pair consists of 2 students, we need to calculate the number of pairs that contain either Jason and Erik, Jason and Jamie, or Erik and Jamie.
The number of pairs that contain Jason and Erik can be calculated using combinations as:
C(2, 2) = 2! / (2!(2 - 2)!) = 2! / (2!0!) = (2 * 1) / (2 * 1) = 1
Similarly, the number of pairs that contain Jason and Jamie or Erik and Jamie is also 1.
So, the total number of pairs that consist of two of the three friends is: 1 + 1 + 1 = 3.
Finally, we can calculate the probability by dividing the number of favorable outcomes (pairs that consist of two of the three friends) by the total number of possible outcomes (total number of pairs):
Probability = Number of favorable outcomes / Total number of possible outcomes = 3 / 210.
Simplifying this fraction, we get:
Probability = 1/70
Therefore, the correct answer is B. 1/70.
To find the probability that two of the three friends will be chosen, we need to determine the total number of possible pairs of students that can be chosen and the number of pairs that include two of the three friends.
First, let's calculate the total number of possible pairs. This can be done using the combination formula, which is nCr = n! / r!(n-r)!. In this case, we have 21 students and we need to choose 2 of them, so the total number of possible pairs is given by:
nCr = 21! / 2!(21-2)!
= 21! / 2!19!
= (21 * 20) / 2
= 420 / 2
= 210
Now, let's calculate the number of pairs that include two of the three friends. Since we want to choose two friends out of three, we can use the combination formula again:
nCr = 3! / 2!(3-2)!
= 3! / 2!1!
= (3 * 2) / 2
= 6 / 2
= 3
So, there are 3 pairs that include two of the three friends.
Finally, to find the probability, we divide the number of favorable outcomes (3) by the total number of possible outcomes (210):
Probability = Number of favorable outcomes / Total number of possible outcomes
= 3 / 210
= 1 / 70
Therefore, the probability that two of the three friends will be chosen is 1/70.
The correct answer is B. 1/70.