Question

kenji and mary are members of a school committee that will be meeting this afternoon. the 6 members of the committee will be seated randomly around a circular table. what is the probability that kenji and mary will not sit next to each other at the meeting? o f. \\(\frac{1}{5}\\) o g. \\(\frac{1}{3}\\) o h. \\(\frac{3}{5}\\) o j. \\(\frac{4}{5}\\)

Explanation:

Step1: Find total circular permutations

For \( n \) people in circular permutation, the number of arrangements is \( (n - 1)! \). Here, \( n = 6 \), so total permutations \( = (6 - 1)! = 5! = 120 \).

Step2: Find permutations where Kenji and Mary sit together

Treat Kenji and Mary as a single entity. Now, we have \( 5 \) entities (the Kenji - Mary pair + 4 others) in circular permutation. The number of arrangements for these \( 5 \) entities is \( (5 - 1)! = 4! \). Also, Kenji and Mary can be arranged within their pair in \( 2! \) ways. So, permutations with them together \( = 4! \times 2! = 24\times2 = 48 \).

Step3: Find probability they sit together

Probability \( P(\text{together})=\frac{\text{Permutations with them together}}{\text{Total permutations}}=\frac{48}{120}=\frac{2}{5} \).

Step4: Find probability they do not sit together

Using the complement rule: \( P(\text{not together}) = 1 - P(\text{together}) = 1-\frac{2}{5}=\frac{3}{5} \).

Answer:

H. \(\frac{3}{5}\)