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 11,880 different ways to appoint the four officers. b. How many different ways can a committee of five be appointed? There are \( \square \) different ways to appoint a committee of five.

To solve part (b), we need to determine how many different ways we can appoint a committee of five members from the 12 qualified candidates. Since the order in which the committee members are selected does not matter, we will use the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of candidates, and \( r \) is the number of members to be selected for the committee. In this case, \( n = 12 \) and \( r = 5 \): \[ \binom{12}{5} = \frac{12!}{5!(12-5)!} = \frac{12!}{5! \cdot 7!} \] Calculating the factorials: \[ 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7! \] So we can simplify: \[ \binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5!} \] Calculating \( 5! \): \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] Now substituting back into the combination formula: \[ \binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{120} \] Calculating the numerator: \[ 12 \times 11 = 132 \] \[ 132 \times 10 = 1320 \] \[ 1320 \times 9 = 11880 \] \[ 11880 \times 8 = 95040 \] Now divide by \( 120 \): \[ \frac{95040}{120} = 792 \] Thus, the number of different ways to appoint a committee of five is: \[ \boxed{792} \]