To determine the apportionment of fourteen legislative seats among the three states using the Huntington-Hill method, we follow these steps:
1. Initial Allocation:
- Allocate to each state the integer part of their standard quota.
- State 1: The standard quota is 2.67. The integer part is 2.
- State 2: The standard quota is 6.92. The integer part is 6.
- State 3: The standard quota is 4.17. The integer part is 4.
2. Calculate Initial Seats:
- State 1: 2 seats
- State 2: 6 seats
- State 3: 4 seats
- Total allocated seats initially: [tex]\(2 + 6 + 4 = 12\)[/tex]
3. Remaining Seats to Allocate:
- Total seats to allocate: 14
- Seats already allocated: 12
- Remaining seats: [tex]\(14 - 12 = 2\)[/tex]
4. Ratios for Huntington-Hill Method:
- Compute the ratios as follows:
- For each state, the ratio is [tex]\(\frac{\text{standard quota}}{\sqrt{\text{seats} \times (\text{seats} + 1)}}\)[/tex].
- State 1: [tex]\(\frac{2.67}{\sqrt{2 \times 3}} = \frac{2.67}{\sqrt{6}}\)[/tex]
- State 2: [tex]\(\frac{6.92}{\sqrt{6 \times 7}} = \frac{6.92}{\sqrt{42}}\)[/tex]
- State 3: [tex]\(\frac{4.17}{\sqrt{4 \times 5}} = \frac{4.17}{\sqrt{20}}\)[/tex]
5. Allocate Remaining Seats:
- We need to allocate the 2 remaining seats to the states with the highest ratios.
- Check the ratios calculated to decide which state gets the next seat.
- Allocate one seat at a time and update the ratios accordingly.
Following this step-by-step procedure, the final allocation of seats results in:
- State 1 ends up with 3 seats.
- State 2 ends up with 7 seats.
- State 3 ends up with 4 seats.
Thus, the apportionment is [tex]\((3, 7, 4)\)[/tex].
This matches option (C) in the provided options:
(C) [tex]\(3, 7, 4)
So, the correct answer is:
(C) \(3, 7, 4\)[/tex]
