Out of the following four choices which does not show the coinciding of the hour hand and minute-hand :

Options

A. `3:16:2`
B. `6:32:43`
C. `9:59:05`
D. `5:27:16`

Answer

To determine which of the given times does not show the coinciding of the hour hand and minute hand, we will use the formula for the overlap of the hour and minute hands:

**Formula:**
\[ \text{Min} = \frac{2}{11} \times t \times 30 \pm \theta \]
Where:
- \( t \) is the hour (in hours)
- \( \theta \) is the angle difference between the two hands (0 degrees when they coincide)

### Step-by-Step Solution:

1. **Option 1: 3:16:02**
- Here, \( t = 3 \) hours.
- Using the formula:
\[ \text{Min} = \frac{2}{11} \times 3 \times 30 \]
- Calculate:
\[ \text{Min} = \frac{2}{11} \times 90 = \frac{180}{11} \approx 16.36 \text{ minutes} \]
- Convert 0.36 minutes to seconds:
\[ 0.36 \times 60 \approx 21.6 \text{ seconds} \approx 22 \text{ seconds} \]
- Therefore, the time of overlap is approximately 3:16:22, which is close to 3:16:02. This option coincides.

2. **Option 2: 6:32:43**
- Here, \( t = 6 \) hours.
- Using the formula:
\[ \text{Min} = \frac{2}{11} \times 6 \times 30 \]
- Calculate:
\[ \text{Min} = \frac{2}{11} \times 180 = \frac{360}{11} \approx 32.73 \text{ minutes} \]
- Convert 0.73 minutes to seconds:
\[ 0.73 \times 60 \approx 43.8 \text{ seconds} \approx 44 \text{ seconds} \]
- Therefore, the time of overlap is approximately 6:32:44, which is close to 6:32:43. This option coincides.

3. **Option 3: 9:59:05**
- Here, \( t = 9 \) hours.
- Using the formula:
\[ \text{Min} = \frac{2}{11} \times 9 \times 30 \]
- Calculate:
\[ \text{Min} = \frac{2}{11} \times 270 = \frac{540}{11} \approx 49.09 \text{ minutes} \]
- Convert 0.09 minutes to seconds:
\[ 0.09 \times 60 \approx 5.4 \text{ seconds} \approx 5 \text{ seconds} \]
- Therefore, the time of overlap is approximately 9:49:05, which is not close to 9:59:05. This option does not coincide.

4. **Option 4: 5:27:16**
- Here, \( t = 5 \) hours.
- Using the formula:
\[ \text{Min} = \frac{2}{11} \times 5 \times 30 \]
- Calculate:
\[ \text{Min} = \frac{2}{11} \times 150 = \frac{300}{11} \approx 27.27 \text{ minutes} \]
- Convert 0.27 minutes to seconds:
\[ 0.27 \times 60 \approx 16.2 \text{ seconds} \approx 16 \text{ seconds} \]
- Therefore, the time of overlap is approximately 5:27:16, which matches exactly. This option coincides.

### Conclusion:
The option that does not show the coinciding of the hour hand and minute hand is **Option 3: 9:59:05**.