Rogers Carrillo
02/16/2023 · Middle School
A corporation must appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO). It must also appoint a planning committee with five different members. There are 12 qualified candidates, and officers can also serve on the committee. Complete parts (a) through (c) below. a. How many different ways can the four officers be appointed? There are different ways to appoint the four officers.
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Step-by-step Solution
To determine the number of different ways to appoint the four officers (president, CEO, COO, and CFO) from 12 qualified candidates, we can use the concept of permutations since the order in which the officers are appointed matters.
The number of ways to choose and arrange \( r \) officers from \( n \) candidates is given by the formula for permutations:
\[
P(n, r) = \frac{n!}{(n - r)!}
\]
In this case, \( n = 12 \) (the total number of candidates) and \( r = 4 \) (the number of officers to be appointed).
Calculating \( P(12, 4) \):
\[
P(12, 4) = \frac{12!}{(12 - 4)!} = \frac{12!}{8!}
\]
This simplifies to:
\[
P(12, 4) = 12 \times 11 \times 10 \times 9
\]
Now, calculating that step-by-step:
1. \( 12 \times 11 = 132 \)
2. \( 132 \times 10 = 1320 \)
3. \( 1320 \times 9 = 11880 \)
Thus, the number of different ways to appoint the four officers is:
\[
\boxed{11880}
\]
Quick Answer
There are 11,880 different ways to appoint the four officers.
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