Explanation:
To understand the degeneracy of a 3-D Simple Harmonic Oscillator (S.H.O) having energy 9/2hv, let's break down the components and concepts involved.

1. Simple Harmonic Oscillator (S.H.O):
A simple harmonic oscillator is a system that exhibits periodic motion around a stable equilibrium position. In the case of a 3-D S.H.O, the oscillator can move in three mutually perpendicular directions, which we can refer to as x, y, and z.

2. Energy Levels of a 3-D S.H.O:
The energy levels of a 3-D S.H.O are quantized, meaning they can only take certain discrete values. The energy of a 3-D S.H.O is given by the equation E = (nx + ny + nz + 3/2)hv, where nx, ny, and nz are the quantum numbers along the x, y, and z directions, respectively. hv represents the energy quantum, which is the product of Planck's constant (h) and the frequency (v) of the oscillator.

3. Determining the Degeneracy:
Degeneracy refers to the number of different quantum states that have the same energy level. To determine the degeneracy of a specific energy level, we need to find the number of different combinations of quantum numbers (nx, ny, nz) that satisfy the energy equation.

4. Energy Level with E = 9/2hv:
Given that the energy of the system is 9/2hv, we can substitute this value into the energy equation and rearrange it to solve for the quantum numbers.

9/2hv = nx + ny + nz + 3/2

Multiplying both sides by 2/3hv, we get:

3/2hv = nx + ny + nz

Since the quantum numbers nx, ny, and nz must be non-negative integers, we can proceed to find all the possible combinations that satisfy the equation.

5. Combinations of Quantum Numbers:
To find the combinations, we can list all the possible values of nx, ny, and nz that satisfy the equation 3/2hv = nx + ny + nz.

- (0, 0, 3)
- (0, 1, 2)
- (0, 2, 1)
- (0, 3, 0)
- (1, 0, 2)
- (1, 1, 1)
- (1, 2, 0)
- (2, 0, 1)
- (2, 1, 0)
- (3, 0, 0)

6. Counting the Combinations:
By counting the number of combinations, we can determine the degeneracy of the energy level.

In this case, there are 10 different combinations, which means the degeneracy of the energy level 9/2hv is 10.

Conclusion:
The degeneracy of a 3-D S.H.O having energy 9/2hv is 10. This means there are 10 different quantum states that have the same energy level.