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?

Explanation:

Step1: Calculate total number of students

Total number of students \(n=8 + 12+4=24\)
We use the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 24\) and \(r = 10\)

$$C(24,10)=\frac{24!}{10!(24 - 10)!}=\frac{24!}{10!×14!}=\frac{24\times23\times22\times21\times20\times19\times18\times17\times16\times15}{10\times9\times8\times7\times6\times5\times4\times3\times2\times1}=1961256$$

Step2: Calculate the number of ways to choose 5 seniors, 3 juniors and 2 sophomores

Number of ways to choose 5 seniors out of 8: \(C(8,5)=\frac{8!}{5!(8 - 5)!}=\frac{8\times7\times6}{3\times2\times1}=56\)
Number of ways to choose 3 juniors out of 12: \(C(12,3)=\frac{12!}{3!(12 - 3)!}=\frac{12\times11\times10}{3\times2\times1}=220\)
Number of ways to choose 2 sophomores out of 4: \(C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4\times3}{2\times1}=6\)
By the multiplication principle, the number of ways to choose 5 seniors, 3 juniors and 2 sophomores is \(N = C(8,5)\times C(12,3)\times C(4,2)=56\times220\times6 = 73920\)

Step3: Calculate the probability

Probability \(P=\frac{73920}{1961256}\approx0.038\)

Answer:

The total number of ways to choose ten mentors is \(1961256\). The probability that the mentors will be five seniors, three juniors, and two sophomores is approximately \(0.038\)