At the first tri-city meeting, there are 8 people from town A, 7 people from town B, and 5 people from town C. If a council consisting of 5 people is randomly selected, find the probability that 3 are from town A and 2 are from town B.
How would I show my work on this equation?
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To find the probability of selecting 3 people from town A and 2 people from town B out of a council of 5 people, you need to calculate the number of favorable outcomes (ways to select 3 people from town A and 2 people from town B) divided by the number of total outcomes (ways to select any 5 people from the entire group).
To calculate the number of favorable outcomes:
1. First, calculate the number of ways to select 3 people from town A out of the available 8 people. This can be done using the combination formula: nCr = n! / (r!(n - r)!), where n is the total number of options, and r is the number of selections. In this case, it is 8C3 = 8! / (3! * (8-3)!).
2. Next, calculate the number of ways to select 2 people from town B out of the available 7 people. Again, use the combination formula: 7C2 = 7! / (2! * (7-2)!).
Now, to calculate the number of total outcomes:
1. Determine the total number of people from which the council of 5 people can be selected. In this case, the total number of people is the sum of people from town A, town B, and town C, which is 8 + 7 + 5 = 20.
2. Use the combination formula to find the number of ways to select any 5 people from the total group of 20 people: 20C5 = 20! / (5! * (20-5)!).
Finally, to calculate the probability, divide the number of favorable outcomes by the number of total outcomes:
Probability = (8C3 * 7C2) / 20C5
To show your work in calculating the final probability, substitute the values into the formulas and perform the necessary calculations. Once you have the values for the combinations, divide the numerator by the denominator to get the final probability.