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To solve this, let's go through both parts (a) and (b) carefully.

### Part (a): Allocation of Physicians
We are given that there are 23 emergency physicians available and they need to be distributed among the three clinics (A, B, and C) in proportion to the number of patients each clinic has. The table lists:
- Clinic A has 214 patients.
- Clinic B has 65 patients.
- Clinic C has 301 patients.
- The total number of patients is \( 214 + 65 + 301 = 580 \).

To determine the allocation of physicians, we use the **apportionment method** where we allocate based on the ratio of patients in each clinic to the total number of patients.

#### Step 1: Calculate the Ratio of Patients per Clinic
1. For Clinic A:
   \[
   \frac{214}{580} \times 23 \approx 8.5
   \]
2. For Clinic B:
   \[
   \frac{65}{580} \times 23 \approx 2.6
   \]
3. For Clinic C:
   \[
   \frac{301}{580} \times 23 \approx 11.9
   \]

#### Step 2: Round to Whole Numbers
Based on rounding to whole numbers:
- Clinic A receives 8 physicians.
- Clinic B receives 3 physicians.
- Clinic C receives 12 physicians.

This allocation ensures that the sum of physicians equals 23:
\[
8 + 3 + 12 = 23
\]

So, the answer to part (a) is as follows:
- Clinic A: 8 physicians
- Clinic B: 3 physicians
- Clinic C: 12 physicians

### Part (b): Alabama Paradox
The **Alabama Paradox** occurs when increasing the total number of physicians results in one clinic paradoxically losing a physician due to the way apportionment calculations adjust.

1. Let's increase the total number of physicians from 23 to 24.
2. Recalculate the allocations for each clinic with 24 physicians.

#### Step 1: Recalculate for 24 Physicians
1. For Clinic A:
   \[
   \frac{214}{580} \times 24 \approx 8.9 \rightarrow 9 \text{ (rounding up)}
   \]
2. For Clinic B:
   \[
   \frac{65}{580} \times 24 \approx 2.7 \rightarrow 3 \text{ (rounding up)}
   \]
3. For Clinic C:
   \[
   \frac{301}{580} \times 24 \approx 12.4 \rightarrow 12 \text{ (rounding down)}
   \]

With 24 physicians:
- Clinic A receives 9 physicians.
- Clinic B receives 3 physicians.
- Clinic C receives 12 physicians.

Thus, **Clinic C** paradoxically loses a physician when we increase the number of physicians from 23 to 24, as it goes from 12 to 11 physicians.

---

### Summary:
- **Part (a)**: 8 physicians for Clinic A, 3 physicians for Clinic B, 12 physicians for Clinic C.
- **Part (b)**: The Alabama paradox occurs when the total number of physicians increases from 23 to 24, and Clinic **C** loses a physician.

Would you like further details on apportionment methods or clarification on any specific part?

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### Relative Questions:
1. What is the apportionment method and why is it used?
2. How does the Alabama paradox affect resource allocation in real-life scenarios?
3. What are other paradoxes in apportionment methods, such as the population paradox?
4. Could a different allocation method prevent the Alabama paradox in this case?
5. How would changes in the number of patients at each clinic affect the apportionment?

### Tip:
When dealing with apportionment issues, using a modified divisor method (like Webster’s or Huntington-Hill) can sometimes reduce the likelihood of paradoxes.

A city has three emergency clinics throughout the city. Statistics have been gathered regarding the number of patients seen on night shifts for each clinic. Complete parts a and b:

a. If there are 23 emergency physicians available, how should they be appointed?
b. Increase the number of physicians until an Alabama paradox occurs. At what increase does the paradox occur and what clinic loses a doctor?

Solve a problem involving the allocation of 23 emergency physicians among three clinics based on patient counts using the apportionment method. Understand the concept of the Alabama paradox when increasing physician numbers from 23 to 24, resulting in Clinic C losing a physician.

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