Final answer:
To determine the number of ways officers can be appointed in a corporation, we use permutations for each position, resulting in n × (n-1) × (n-2) × (n-3) possibilities. The planning committee of five is chosen with combinations, calculated by C(n, 5) = n! / [5!(n-5)!].
Step-by-step explanation:
Calculating Different Ways to Appoint Officers
To find out the number of different ways the officers (president, CEO, COO, and CFO) can be appointed, we use the principle of permutations, because the order in which we select the officers matters. Assuming we have a different person for each role and there are n candidates in total, the first officer can be appointed in n ways, the second in (n-1) ways, the third in (n-2) ways, and the fourth in (n-3) ways. Therefore, the total number of different ways the officers can be appointed is n × (n-1) × (n-2) × (n-3).
Choosing a Planning Committee
When it comes to the planning committee with five members, we are dealing with combinations, as the order of selection does not matter. If there are n candidates left after choosing the officers, then the number of ways to choose the committee is given by the combination formula: C(n, 5) = \frac{n!}{5!(n-5)!}, where '!' denotes factorial. Therefore, with n candidates, the process involves selecting officers first and then the committee members.