A city council consists of eight Democrats and seven Republicans. If a committee of six people is​ selected, find the probability of selecting three Democrats and three Republicans.

To find the probability of selecting three Democrats and three Republicans, we need to determine the number of ways we can choose three Democrats out of eight and three Republicans out of seven, and divide it by the total number of ways we can choose any six people out of fifteen.

The number of ways to choose three Democrats out of eight is denoted as "C(8, 3)" or "8 choose 3." The formula for combinations is:

C(n, k) = n! / (k!(n-k)!)

Using this formula, we can calculate:

C(8, 3) = 8! / (3!(8-3)!)
= 8! / (3!5!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56

Similarly, the number of ways to choose three Republicans out of seven is:

C(7, 3) = 7! / (3!(7-3)!)
= 7! / (3!4!)
= (7 * 6 * 5) / (3 * 2 * 1)
= 35

Now we need to calculate the total number of ways to choose any six people out of fifteen:

C(15, 6) = 15! / (6!(15-6)!)
= 15! / (6!9!)
= (15 * 14 * 13 * 12 * 11 * 10) / (6 * 5 * 4 * 3 * 2 * 1)
= 5005

Finally, we can calculate the probability:

Probability = (Number of ways to choose 3 Democrats) * (Number of ways to choose 3 Republicans) / (Number of ways to choose any 6 people)
= (56 * 35) / 5005
= 1960 / 5005
≈ 0.3912

Therefore, the probability of selecting three Democrats and three Republicans is approximately 0.3912, or 39.12%.

To find the probability of selecting three Democrats and three Republicans from a committee of six people, we need to determine the total number of possible committees and the number of committees that have three Democrats and three Republicans.

To calculate the total number of possible committees, we need to choose six people out of the total number of council members (which is 8 Democrats + 7 Republicans = 15). This can be calculated using combination formula, denoted as C(n, k), which represents the number of ways to choose k items out of a total of n items:

C(15, 6) = 15! / (6! * (15-6)!) = 5005

Next, we need to determine the number of committees that have three Democrats and three Republicans.

We can do this by calculating the number of ways to choose 3 Democrats out of 8 Democrats and the number of ways to choose 3 Republicans out of 7 Republicans. Then, we multiply these two values together to get the total number of committees with three Democrats and three Republicans.

C(8, 3) = 8! / (3! * (8-3)!) = 56

C(7, 3) = 7! / (3! * (7-3)!) = 35

Total number of committees with three Democrats and three Republicans = C(8, 3) * C(7, 3) = 56 * 35 = 1,960

Finally, we can find the probability by dividing the number of committees with three Democrats and three Republicans by the total number of possible committees:

Probability = Number of favorable outcomes / Total number of outcomes

Probability = 1,960 / 5005 ≈ 0.391 or 39.1%

Therefore, the probability of selecting three Democrats and three Republicans from the committee of six people is approximately 0.391 or 39.1%.

C(8,3) x C(7,3)

= 56(35)
= 1960