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?
15·10·6
5005
900
5005
180
1001
the probability of choosing 2 seniors, 2 juniors, and 2 sophomores is 180/1001.
what is the probability that the students elect 3 seniors, 2 juniors, and 1 sophomore?
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 total number of ways to choose 6 students

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, $_{15}C_{6}=\frac{15!}{6!(15 - 6)!}=\frac{15!}{6!9!}=\frac{15\times14\times13\times12\times11\times10}{6\times5\times4\times3\times2\times1}=5005$.

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 $_{6}C_{2}=\frac{6!}{2!(6 - 2)!}=\frac{6!}{2!4!}=\frac{6\times5}{2\times1}=15$.
The number of ways to choose 2 juniors out of 5 is $_{5}C_{2}=\frac{5!}{2!(5 - 2)!}=\frac{5!}{2!3!}=\frac{5\times4}{2\times1}=10$.
The number of ways to choose 2 sophomores out of 4 is $_{4}C_{2}=\frac{4!}{2!(4 - 2)!}=\frac{4!}{2!2!}=\frac{4\times3}{2\times1}=6$.
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 the number of favorable outcomes divided by the number of total outcomes. So $P=\frac{_{6}C_{2}\times_{5}C_{2}\times_{4}C_{2}}{_{15}C_{6}}=\frac{900}{5005}=\frac{180}{1001}$.

Answer:

$\frac{180}{1001}$