#### Answer

(a) Using Hamilton's method, each state is apportioned the following number of seats:
State A is apportioned 2 seats.
State B is apportioned 18 seats.
State C is apportioned 20 seats.
(b) The Alabama paradox occurs. Initially, with a total number of 40 seats, State A was allocated 2 seats. After the total number of seats increased to 41, State A was allocated only 1 seat.

#### Work Step by Step

(a) We can find the standard divisor.
$standard~divisor = \frac{total ~population}{number~of~ seats}$
$standard~divisor = \frac{20,000}{40}$
$standard~divisor = 500$
We can find each state's standard quota. The standard quota of each state is the state's population divided by the standard divisor.
State A:
$standard ~quota = \frac{population}{standard~divisor}$
$standard~quota = \frac{680}{500}$
$standard~quota = 1.36$
State B:
$standard ~quota = \frac{population}{standard~divisor}$
$standard~quota = \frac{9150}{500}$
$standard~quota = 18.3$
State C:
$standard ~quota = \frac{population}{standard~divisor}$
$standard~quota = \frac{10,170}{500}$
$standard~quota = 20.34$
Hamilton's method is an apportionment method that involves rounding each standard quota down to the nearest whole number. Surplus seats are given, one at a time, to the states with the largest decimal parts in their standard quotas until there are no more surplus seats.
Initially, each state is apportioned its lower quota.
State A is apportioned 1 seat.
State B is apportioned 18 seats.
State C is apportioned 20 seats.
The total number of seats which have been apportioned is 1 + 18 + 20 = 39 seats
Since there is a total of 40 seats, there is one surplus seat. One more seat is given to State A because it has the largest decimal part (0.36) in its standard quota.
Using Hamilton's method, each state is apportioned the following number of seats:
State A is apportioned 1 + 1 = 2 seats.
State B is apportioned 18 seats.
State C is apportioned 20 seats.
(b) We can find the standard divisor.
$standard~divisor = \frac{total ~population}{number~of~ seats}$
$standard~divisor = \frac{20,000}{41}$
$standard~divisor = 487.8$
We can find each state's standard quota. The standard quota of each state is the state's population divided by the standard divisor.
State A:
$standard ~quota = \frac{population}{standard~divisor}$
$standard~quota = \frac{680}{487.8}$
$standard~quota = 1.39$
State B:
$standard ~quota = \frac{population}{standard~divisor}$
$standard~quota = \frac{9150}{487.8}$
$standard~quota = 18.76$
State C:
$standard ~quota = \frac{population}{standard~divisor}$
$standard~quota = \frac{10,170}{487.8}$
$standard~quota = 20.85$
Hamilton's method is an apportionment method that involves rounding each standard quota down to the nearest whole number. Surplus seats are given, one at a time, to the states with the largest decimal parts in their standard quotas until there are no more surplus seats.
Initially, each state is apportioned its lower quota.
State A is apportioned 1 seat.
State B is apportioned 18 seats.
State C is apportioned 20 seats.
The total number of seats which have been apportioned is 1 + 18 + 20 = 39 seats
Since there is a total of 41 seats, there are two surplus seats. The first seat is given to State C because it has the largest decimal part (0.85) in its standard quota. The second seat is given to State B because it has the second largest decimal part (0.76) in its standard quota.
Using Hamilton's method, each state is apportioned the following number of seats:
State A is apportioned 1 seat.
State B is apportioned 18 + 1 = 19 seats.
State C is apportioned 20 + 1 = 21 seats.
We can see that the Alabama paradox occurs. Initially, with 40 seats, State A was allocated 2 seats. After the number of seats increased to 41, State A was allocated only 1 seat. Therefore, the Alabama paradox occurs.