Claudia is arranging seating for her sorority's fall banquet. She has to sit 7 seniors, 5 juniors, 9 sophomores and 3 freshmen at the end banquet table. What is the probability members of the same class will all sit together?
7!x5!x9!x3!= 1.317x10^12
Am I correct?
first of all, they asked for a probability.
How many ways to seat all the people?
Now, consider all the members of a class as one person. How many ways to seat them now?
Do the same for the other classes.
(Total ways to seat fused classes) divided by (total ways to seat everyone) is the probability.
To determine the probability of members of the same class sitting together, we need to calculate the total number of possible seat arrangements where members of the same class are together and divide it by the total number of possible seat arrangements.
First, let's consider the seniors. There are 7 seniors, and they need to sit together. We can treat this group as a single unit, so there are 6! (6 factorial) ways to arrange the seniors within their group.
Next, let's consider the juniors. There are 5 juniors, and they need to sit together. Similarly, we can treat this group as a single unit, so there are 4! ways to arrange the juniors within their group.
Following the same logic, we can treat the sophomores as a single unit, so there are 8! ways to arrange them.
Lastly, the freshmen need to sit together, so we treat them as a unit, giving us 2! ways to arrange them.
To find the total number of seat arrangements where members of the same class are together, we need to multiply these values together:
6! × 4! × 8! × 2! = 345,600
Now, let's find the total number of possible seat arrangements without any restrictions. There are a total of 24 members in the sorority, so there are 24! ways to arrange them.
To find the probability, we divide the number of favorable outcomes (where members of the same class are together) by the total number of outcomes:
Probability = (345,600 / 24!) ≈ 0.04575
Therefore, the probability that members of the same class will sit together is approximately 0.04575 or 4.575%.
The calculation you provided (7! × 5! × 9! × 3!) is incorrect. It seems like you multiplied the factorials together without considering the groups properly.