A group meeting is attended by 14 delegates. 9 of the delegates are from Atlantis and 5 of the
delegates are from Zedonia.
(a) In how many ways can the group choose 6 of the 14 delegates to form a negotiation panel?
Show work.
(b) In how many ways can the group choose 6 of the 14 delegates to form the negotiation panel,
if 3 delegates must be from Atlantis and 3 delegates must be from Zedonia? Show work.
(c) If a 6-person negotiation panel is selected at random from the 14 delegates, what is the
probability the negotiation panel consists of 3 delegates from Atlantis and 3 delegates from
Zedonia? Show work.
(a) To answer this question, we can use the concept of combinations. Since the order of selection does not matter, we need to find the number of ways to choose 6 delegates out of 14. The formula for combinations is given by:
C(n, r) = n! / (r!(n-r)!)
where n is the total number of delegates (14 in this case), and r is the number of delegates we want to choose (6 in this case).
Plugging in the values, we get:
C(14, 6) = 14! / (6!(14-6)!)
= 3003
Therefore, there are 3003 ways to choose a 6-member negotiation panel from a group of 14 delegates.
(b) In this case, we need to choose 3 delegates from Atlantis and 3 delegates from Zedonia. We can calculate this by finding the product of the number of ways to choose 3 delegates from Atlantis and 3 delegates from Zedonia.
The number of ways to choose 3 delegates from Atlantis can be calculated using combinations:
C(9, 3) = 9! / (3!(9-3)!)
= 84
Similarly, the number of ways to choose 3 delegates from Zedonia can be calculated using combinations:
C(5, 3) = 5! / (3!(5-3)!)
= 10
To get the total number of ways, we multiply the two results:
Total number of ways = C(9, 3) * C(5, 3)
= 84 * 10
= 840
Therefore, there are 840 ways to choose a 6-member negotiation panel with 3 delegates from Atlantis and 3 delegates from Zedonia.
(c) To calculate the probability, we need to find the number of favorable outcomes (selecting 3 delegates from Atlantis and 3 delegates from Zedonia) and divide it by the total number of possible outcomes.
The number of ways to choose 3 delegates from Atlantis can be calculated using combinations:
C(9, 3) = 9! / (3!(9-3)!)
= 84
Similarly, the number of ways to choose 3 delegates from Zedonia can be calculated using combinations:
C(5, 3) = 5! / (3!(5-3)!)
= 10
To get the total number of possible outcomes, we use the result from part (a):
Total number of possible outcomes = C(14, 6)
= 3003
Therefore, the probability is:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= (C(9, 3) * C(5, 3)) / C(14, 6)
= (84 * 10) / 3003
= 840 / 3003
≈ 0.2794
Therefore, the probability that a 6-person negotiation panel consists of 3 delegates from Atlantis and 3 delegates from Zedonia is approximately 0.2794.