A city council consists of five democrats and five republicans. If a committee of seven people is selected, find the probability of selecting three Democrats and four republicans.
The total number of ways to choose a committee of 7 people from a group of 10 (5 Democrats and 5 Republicans) is given by the combination formula:
C(10, 7) = 10! / (7! * (10-7)!) = 10! / (7! * 3!) = (10 * 9 * 8) / (3 * 2 * 1) = 120.
The number of ways to select exactly 3 Democrats and 4 Republicans is the product of the number of ways to choose 3 Democrats from the 5 available and the number of ways to choose 4 Republicans from the 5 available:
C(5, 3) * C(5, 4) = (5! / (3! * (5-3)!)) * (5! / (4! * (5-4)!)) = (5 * 4) * (5 * 1) = 20 * 5 = 100.
Therefore, the probability of selecting three Democrats and four Republicans on the committee is:
100 / 120 = 5/6.