Question 1200459: 3. A corporation must appoint a president, chief executive officer (CEO), chief financial officer (CFO) and chief operating officer (COO). It must also appoint a planning committee with six different members. There are 12 qualified candidates, and officers can also serve on the committee.
(i) How many different ways can the officers be appointed?
(ii) What is the probability of randomly selecting the officers and getting the youngest candidate a president?
(iii) How many different ways can the committee be appointed?
(iv) What is the probability of randomly selecting the committee members and getting the six youngest of the qualified candidates?
Answer by GingerAle(43)   (Show Source):
You can put this solution on YOUR website! Certainly, let's break down the calculations for the corporation's appointments.
**i) How many different ways can the officers be appointed?**
* This is a permutation problem since the order of the officers (President, CEO, CFO, COO) matters.
* We have 12 candidates and need to choose 4 for the officer positions.
* The formula for permutations is:
P(n, r) = n! / (n-r)!
where:
* n is the total number of items (candidates)
* r is the number of items to choose (officers)
* P(12, 4) = 12! / (12-4)! = 12! / 8! = 12 * 11 * 10 * 9 = 11,880
* **There are 11,880 different ways to appoint the officers.**
**ii) What is the probability of randomly selecting the officers and getting the youngest candidate a president?**
* There is only 1 youngest candidate.
* The probability of selecting the youngest candidate as president is 1 out of the total number of possible presidents (which is the total number of candidates).
* Probability = 1 / 12 = 0.0833 (approximately)
* **The probability of randomly selecting the officers and getting the youngest candidate as president is approximately 0.0833 or 8.33%.**
**iii) How many different ways can the committee be appointed?**
* This is a combination problem since the order of the committee members does not matter.
* We have 12 candidates and need to choose 6 for the committee.
* The formula for combinations is:
C(n, r) = n! / (r! * (n-r)!)
where:
* n is the total number of items (candidates)
* r is the number of items to choose (committee members)
* C(12, 6) = 12! / (6! * (12-6)!) = 12! / (6! * 6!) = 924
* **There are 924 different ways to appoint the committee.**
**iv) What is the probability of randomly selecting the committee members and getting the six youngest of the qualified candidates?**
* There is only 1 possible combination of the six youngest candidates.
* Probability = 1 / 924 = 0.001082 (approximately)
* **The probability of randomly selecting the committee members and getting the six youngest of the qualified candidates is approximately 0.001082 or 0.1082%.**
Let me know if you'd like to explore any other scenarios or have further questions!
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