Question

a group of 6 seniors, 5 juniors, and 4 sophomores run for student council. assume that each student has an equal chance of being elected to student council. determine each probability and express your answers as fractions in lowest terms. sample problem what is the probability that the students elect 2 seniors, 2 juniors, and 2 sophomores? ( \frac{_{6}c_{2}cdot_{5}c_{2}cdot_{4}c_{2}}{_{15}c_{6}}=\frac{900}{5005}=\frac{180}{1001} ) what is the probability that the students elect 2 seniors and 4 sophomores? enter the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Calculate total number of ways to choose 6 students

The total number of students is \(6 + 5+4=15\). We want to choose 6 students out of 15. Using the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 15\) and \(r=6\), we have \(C(15,6)=\frac{15!}{6!(15 - 6)!}=\frac{15!}{6!9!}=5005\).

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

There are 6 seniors, and we want to choose 2, so \(C(6,2)=\frac{6!}{2!(6 - 2)!}=\frac{6!}{2!4!}=15\). There are 5 juniors, and we want to choose 2, so \(C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5!}{2!3!}=10\). There are 4 sophomores, and we want to choose 2, so \(C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4!}{2!2!}=6\). The number of ways to choose 2 seniors, 2 juniors and 2 sophomores is \(C(6,2)\times C(5,2)\times C(4,2)=15\times10\times6 = 900\).

Step3: Calculate the probability

The probability \(P\) is the number of favorable outcomes divided by the number of total outcomes. So \(P=\frac{900}{5005}=\frac{180}{1001}\).

Answer:

\(\frac{180}{1001}\)