Question 1201073
Certainly, let's break down the calculations for each part:

**i) How many different ways can the officers be appointed?**

* We have 12 candidates for 4 positions: President, CEO, CFO, and COO. 
* The order of selection matters. 
* This is a permutation problem.

* Number of ways to appoint officers = 12P4 = 12! / (12-4)! = 12! / 8! = 12 * 11 * 10 * 9 = 11,880 ways

**ii) What is the probability of randomly selecting the officers and getting the youngest candidate a president?**

* There is only 1 youngest candidate.
* For the President position, there is 1 favorable outcome (youngest candidate) out of 12 possible candidates.
* For the other 3 positions (CEO, CFO, COO), there are no restrictions.

* Probability = (1/12) * (11/11) * (10/10) * (9/9) = 1/12

**iii) How many different ways can the committee be appointed?**

* We need to select 6 members out of 12 candidates.
* The order of selection does not matter.
* This is a combination problem.

* Number of ways to appoint the committee = 12C6 = 12! / (6! * (12-6)!) = 12! / (6! * 6!) = 924 ways

**iv) What is the probability of randomly selecting the committee members and getting the six youngest of the qualified candidates?**

* There is only 1 favorable outcome (selecting the 6 youngest candidates) out of 924 possible combinations.

* Probability = 1/924

Let me know if you have any other questions or would like to explore different scenarios!