If the subsidiary quantum number of a subenergy level is `4`, the maximum and minimum values of the spin multiplicities are:

Options

A. `9, 1`
B. `10, 1`
C. `10, 2`
D. `4, -4`

Explanation

To solve the problem regarding the maximum and minimum values of spin multiplicities when the subsidiary quantum number (L) is given as 4, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Subsidiary Quantum Number (L)**:
- We are given that the subsidiary quantum number (L) is 4.

2. **Calculate the Number of Degeneracy of Orbitals**:
- The formula for the number of degeneracy of orbitals is given by:
\[ \text{Degeneracy} = 2L + 1 \]
- Substituting the value of L:
\[ \text{Degeneracy} = 2 \times 4 + 1 = 8 + 1 = 9 \]

3. **Calculate the Maximum Total Spin (S)**:
- The maximum total spin is calculated as:
\[ S = \frac{\text{Degeneracy}}{2} = \frac{9}{2} \]

4. **Calculate the Maximum Spin Multiplicity**:
- The formula for spin multiplicity is:
\[ \text{Multiplicity} = 2S + 1 \]
- Substituting the value of S:
\[ \text{Maximum Multiplicity} = 2 \times \frac{9}{2} + 1 = 9 + 1 = 10 \]

5. **Calculate the Minimum Spin (S)**:
- The minimum spin is considered to be \( \frac{1}{2} \).

6. **Calculate the Minimum Spin Multiplicity**:
- Using the same multiplicity formula:
\[ \text{Minimum Multiplicity} = 2 \times \frac{1}{2} + 1 = 1 + 1 = 2 \]

7. **Final Result**:
- The maximum and minimum values of spin multiplicities are:
\[ \text{Maximum Multiplicity} = 10, \quad \text{Minimum Multiplicity} = 2 \]
- Therefore, the answer is \( (10, 2) \).

### Final Answer:
The maximum and minimum values of spin multiplicities are \( (10, 2) \).

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