Question

a group of 6 seniors, 5 juniors, and 4 sophomores run for student council. the council has 6 members. 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? $\frac{{_6}c_2cdot{_5}c_2cdot{_4}c_2}{{_{15}}c_6}=\frac{15cdot10cdot6}{5005}=\frac{900}{5005}=\frac{180}{1001}$ the probability of choosing 2 seniors, 2 juniors, and 2 sophomores is $\frac{180}{1001}$. what is the probability that the students elect 3 juniors and 3 sophomores? enter the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Calculate total number of students

There are 6 seniors, 5 juniors, and 4 sophomores. So the total number of students is $6 + 5+4=15$.

Step2: Calculate the number of ways to choose 6 students for the council

The number of ways to choose 6 students out of 15 is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 15$ and $r=6$. So, $_{15}C_{6}=\frac{15!}{6!(15 - 6)!}=\frac{15!}{6!9!}=5005$.

Step3: Calculate the number of ways to choose 2 seniors, 2 juniors and 2 sophomores

The number of ways to choose 2 seniors out of 6 is $_{6}C_{2}=\frac{6!}{2!(6 - 2)!}=\frac{6!}{2!4!}=15$.
The number of ways to choose 2 juniors out of 5 is $_{5}C_{2}=\frac{5!}{2!(5 - 2)!}=\frac{5!}{2!3!}=10$.
The number of ways to choose 2 sophomores out of 4 is $_{4}C_{2}=\frac{4!}{2!(4 - 2)!}=\frac{4!}{2!2!}=6$.
By the multiplication - principle, the number of ways to choose 2 seniors, 2 juniors and 2 sophomores is $_{6}C_{2}\times_{5}C_{2}\times_{4}C_{2}=15\times10\times6 = 900$.

Step4: Calculate the probability

The probability $P$ of choosing 2 seniors, 2 juniors and 2 sophomores is $P=\frac{_{6}C_{2}\times_{5}C_{2}\times_{4}C_{2}}{_{15}C_{6}}=\frac{900}{5005}=\frac{180}{1001}$.

Answer:

$\frac{180}{1001}$