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? $\frac{_{6}c_{2}cdot_{5}c_{2}cdot_{4}c_{2}}{_{15}c_{6}}=\frac{15cdot10cdot6}{5005}=\frac{900}{5005}=\frac{180}{1001}$ the probability of choosing 2 seniors, 2 juniors, and 2 sophomores is $\frac{180}{1001}$. what is the probability that the students elect all seniors? 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 $6 + 5+4=15$. The council has 6 members. The total number of ways to choose 6 members out of 15 is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 15$ and $r = 6$. So, $_{15}C_{6}=\frac{15!}{6!(15 - 6)!}=\frac{15!}{6!9!}=\frac{15\times14\times13\times12\times11\times10}{6\times5\times4\times3\times2\times1}=5005$.

Step2: Calculate number of ways to choose all - seniors

The number of ways to choose 6 seniors out of 6 is $_{6}C_{6}=\frac{6!}{6!(6 - 6)!}=1$.

Step3: Calculate the probability

The probability $P$ of choosing all - seniors is the number of favorable outcomes (choosing all - seniors) divided by the number of total outcomes (choosing any 6 students). So $P=\frac{_{6}C_{6}}{_{15}C_{6}}=\frac{1}{5005}$.

Answer:

$\frac{1}{5005}$