Step 1 :First, we need to calculate the total number of ways to fill the 4 offices from the 17 candidates. This is a combination problem, and can be calculated as \(C(17, 4)\), which equals 2380.
Step 2 :For part (a), we need to calculate the probability that all of the offices are filled by members of the debate team. This is a combination problem, where we are choosing 4 people out of 9 (the number of debate team members). The number of ways to do this is \(C(9, 4)\), which equals 126.
Step 3 :The probability for part (a) is then the number of ways to choose 4 debate team members divided by the total number of ways to fill the offices, or \(\frac{126}{2380}\), which equals 0.053 when rounded to three decimal places.
Step 4 :For part (b), we need to calculate the probability that none of the offices are filled by members of the debate team. This is also a combination problem, where we are choosing 4 people out of 8 (the number of non-debate team members). The number of ways to do this is \(C(8, 4)\), which equals 70.
Step 5 :The probability for part (b) is then the number of ways to choose 4 non-debate team members divided by the total number of ways to fill the offices, or \(\frac{70}{2380}\), which equals 0.029 when rounded to three decimal places.
Step 6 :Final Answer: (a) The probability that all of the offices are filled by members of the debate team is \(\boxed{0.053}\) (rounded to three decimal places). (b) The probability that none of the offices are filled by members of the debate team is \(\boxed{0.029}\) (rounded to three decimal places).