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? what is the probability that the students elect all seniors? > enter the answer in the space provided. use numbers instead of words.

Explanation:

Part 1: Probability of electing 2 seniors, 2 juniors, and 2 sophomores

Step 1: Calculate total number of students

There are 6 seniors, 5 juniors, and 4 sophomores. So total number of students \( n = 6 + 5 + 4 = 15 \). We need to choose 6 members for the council. The total number of ways to choose 6 students from 15 is given by the combination formula \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 15 \) and \( k = 6 \). So total number of ways \( C(15, 6)=\frac{15!}{6!(15 - 6)!}=\frac{15!}{6!9!}=\frac{15\times14\times13\times12\times11\times10}{6\times5\times4\times3\times2\times1}=5005 \).

Step 2: Calculate number of favorable ways

We need to choose 2 seniors from 6, 2 juniors from 5, and 2 sophomores from 4.

• Number of ways to choose 2 seniors from 6: \( C(6, 2)=\frac{6!}{2!(6 - 2)!}=\frac{6!}{2!4!}=\frac{6\times5}{2\times1} = 15 \)
• Number of ways to choose 2 juniors from 5: \( C(5, 2)=\frac{5!}{2!(5 - 2)!}=\frac{5!}{2!3!}=\frac{5\times4}{2\times1}=10 \)
• Number of ways to choose 2 sophomores from 4: \( C(4, 2)=\frac{4!}{2!(4 - 2)!}=\frac{4!}{2!2!}=\frac{4\times3}{2\times1} = 6 \)

The number of favorable ways is the product of these three combinations: \( 15\times10\times6 = 900 \).

Step 3: Calculate the probability

Probability \( P=\frac{\text{Number of favorable ways}}{\text{Total number of ways}}=\frac{900}{5005} \). Simplify this fraction by dividing numerator and denominator by 5: \( \frac{900\div5}{5005\div5}=\frac{180}{1001} \).

Part 2: Probability of electing all seniors

Step 1: Total number of ways (already calculated)

Total number of ways to choose 6 students from 15 is \( C(15, 6) = 5005 \) (from part 1).

Step 2: Number of favorable ways (electing all seniors)

We need to choose 6 seniors from 6. The number of ways to do this is \( C(6, 6)=\frac{6!}{6!(6 - 6)!}=\frac{6!}{6!0!}=1 \) (since \( 0! = 1 \)).

Step 3: Calculate the probability

Probability \( P=\frac{\text{Number of favorable ways}}{\text{Total number of ways}}=\frac{1}{5005} \).

Answer:

s:

• Probability of 2 seniors, 2 juniors, 2 sophomores: \(\frac{180}{1001}\)
• Probability of all seniors: \(\frac{1}{5005}\)