Question

Maryvale High School is starting a freshman mentoring program. There are eight seniors, twelve juniors, and four sophomores signed up to mentor. Ten mentors will be chosen for the school year. Round answers to three decimal places. What are the total number of ways to choose ten mentors from the group of seniors, juniors, and sophomores? What is the probability that the mentors will be five seniors, three juniors, and two sophomores?

Answer 1

Explanation:

.038

Answer 2

1) 1961256

2) 0.04

Explanation:

Number of seniors = 8

Number of juniors = 12

Number of sophomores = 4

Total number of students = 8 + 12 + 4 = 24

Part 1) Choosing 10 mentors

This part requires choosing 10 students from a group of 24 students. This is a problem of combinations and 10 students can be chosen from 24 in 24C10 ways, which equals:

`24C10=(24!)/(10!*(24-10)!) \\\\ =(24!)/(10!*14!) \\\\ =1961256`

This means we can chose 10 mentors from the group of seniors, juniors, and sophomores in 1,961,256 ways.

Part 2)

The previous part gave the total possible number of ways to select 10 mentors without any restriction of the selection. In this part we have to chose 5 seniors from 8, 3 juniors from 12 and 2 sophomores from 4.

Number of ways to chose 5 seniors = 8C5 = 56

Number of ways to chose 3 juniors = 12C3 = 220

Number of ways to chose 2 sophomores = 4C2 = 6

Thus, by the fundamental principle of counting the total number of ways of choosing five seniors, three juniors, and two sophomores will be = 56 x 220 x 6 = 73,920

This gives us the number of desired outcomes. The previous part gives us the number of total possible outcomes.

`Probability = \frac{\text{Number of Desired outcomes}}{\text{Total number of outcomes}}\\\\ = (73920)/(1961256) \\\\ =0.04`

Thus, the probability that the mentors will be five seniors, three juniors, and two sophomores is approximately 0.04