#### Answer

When there were 10 doctors, Clinic A was apportioned 3 doctors. After the total number of doctors increased from 10 to 11, Clinic A was only apportioned 2 doctors. Therefore, the Alabama paradox occurs.

#### Work Step by Step

We can find the total patient load.
total load = 119 + 165 + 216 = 500
We can find the standard divisor when there are 10 doctors.
$standard~divisor = \frac{total~load}{number~of~ doctors}$
$standard~divisor = \frac{500}{10}$
$standard~divisor = 50$
The standard divisor is 50.
We can find the standard quota for each clinic.
Clinic A:
$standard~quota = \frac{patient~load}{standard~divisor}$
$standard~quota = \frac{119}{50}$
$standard~quota = 2.38$
Clinic B:
$standard~quota = \frac{patient~load}{standard~divisor}$
$standard~quota = \frac{165}{50}$
$standard~quota = 3.30$
Clinic C:
$standard~quota = \frac{patient~load}{standard~divisor}$
$standard~quota = \frac{216}{50}$
$standard~quota = 4.32$
Hamilton's method is an apportionment method that involves rounding each standard quota down to the nearest whole number. Surplus doctors are given, one at a time, to the clinics with the largest decimal parts in their standard quotas until there are no more surplus doctors.
Initially, each clinic is apportioned its lower quota.
Clinic A is apportioned 2 doctors.
Clinic B is apportioned 3 doctors.
Clinic C is apportioned 4 doctors.
The total number of doctors which have been apportioned is 2 + 3 + 4 = 9 doctors
Since there is a total of 10 doctors, there is one surplus doctor. One more doctor is given to Clinic A because it has the largest decimal part (0.38) in its standard quota.
Using Hamilton's method, each clinic is apportioned the following number of doctors:
Clinic A is apportioned 2 + 1 = 3 doctors.
Clinic B is apportioned 3 doctors.
Clinic C is apportioned 4 doctors.
Let's suppose the number of doctors is increased from 10 to 11. We can find the standard divisor when there are 11 doctors.
$standard~divisor = \frac{total~load}{number~of~ doctors}$
$standard~divisor = \frac{500}{11}$
$standard~divisor = 45.45$
The standard divisor is 45.45.
We can find the standard quota for each clinic.
Clinic A:
$standard~quota = \frac{patient~load}{standard~divisor}$
$standard~quota = \frac{119}{45.45}$
$standard~quota = 2.62$
Clinic B:
$standard~quota = \frac{patient~load}{standard~divisor}$
$standard~quota = \frac{165}{45.45}$
$standard~quota = 3.63$
Clinic C:
$standard~quota = \frac{patient~load}{standard~divisor}$
$standard~quota = \frac{216}{45.45}$
$standard~quota = 4.75$
Hamilton's method is an apportionment method that involves rounding each standard quota down to the nearest whole number. Surplus doctors are given, one at a time, to the clinics with the largest decimal parts in their standard quotas until there are no more surplus doctors.
Initially, each clinic is apportioned its lower quota.
Clinic A is apportioned 2 doctors.
Clinic B is apportioned 3 doctors.
Clinic C is apportioned 4 doctors.
The total number of doctors which have been apportioned is 2 + 3 + 4 = 9 doctors
Since there is a total of 11 doctors, there are two surplus doctors. The first doctor is given to Clinic C because it has the largest decimal part (0.75) in its standard quota. The second doctor is given to Clinic B because it has the second largest decimal part (0.63) in its standard quota.
Using Hamilton's method, each clinic is apportioned the following number of doctors:
Clinic A is apportioned 2 doctors.
Clinic B is apportioned 3 + 1 = 4 doctors.
Clinic C is apportioned 4 + 1 = 5 doctors.
When there were 10 doctors, Clinic A was apportioned 3 doctors. After the total number of doctors increased from 10 to 11, Clinic A was only apportioned 2 doctors. Therefore, the Alabama paradox occurs.