at the first tri city meeting where 8 people are from twon A 7 people are from town B and 5 people are from town C. If the council consists of 5 people find the probability that 3 people are from town A and 2 people are from town B. How do i do this
To find the probability that 3 people are from Town A and 2 people are from Town B, we can follow these steps:
Step 1: Calculate the total number of ways to select 5 people from the total of 8 people from Town A, 7 people from Town B, and 5 people from Town C. This can be done using combinations:
C(8, 3) * C(7, 2) * C(5, 0) = (8! / (3!(8-3)!)) * (7! / (2!(7-2)!)) * (5! / (0!(5-0)!))
Step 2: Calculate the total number of ways to select any 5 people from the total 20 attendees (8 from Town A, 7 from Town B, and 5 from Town C):
C(20, 5) = 20! / (5! * (20-5)!)
Step 3: Divide the number of ways to select 3 people from Town A and 2 people from Town B by the total number of ways to select any 5 people from the total attendees:
P = (C(8, 3) * C(7, 2) * C(5, 0)) / C(20, 5)
Now you can substitute the values and calculate the probability P.
To find the probability of this event, we need to consider the total number of ways the council can be formed and the favorable outcomes where 3 people are from town A and 2 people are from town B.
First, let's calculate the total number of ways the council can be formed. Since there are 8 people from town A, 7 people from town B, and 5 people from town C, we have a total of 8 + 7 + 5 = 20 people to choose from to form the council.
The number of ways to choose 5 people out of 20 can be calculated using the combination formula, also known as "nCr". The formula is:
nCr = n! / (r! * (n - r)!)
where n represents the total number of items to choose from, and r represents the number of items we want to choose.
Using this formula, we can calculate the total number of ways to choose 5 people out of 20:
Total number of ways = 20! / (5! * (20 - 5)!) = 20! / (5! * 15!)
Now, let's consider the favorable outcomes where 3 people are from town A and 2 people are from town B. We need to multiply the number of ways to choose 3 people from town A with the number of ways to choose 2 people from town B:
Number of favorable outcomes = (8! / (3! * (8 - 3)!) * (7! / (2! * (7 - 2)!)
Finally, we can calculate the probability of this event by dividing the number of favorable outcomes by the total number of ways:
Probability = Number of favorable outcomes / Total number of ways
Now, you can plug in the values in the formulas and simplify to find the actual probability.