To solve this problem, we can use a Venn diagram. Let's break down the information given in the problem:

- Half the number of experts voted for Mr. A: This means that the number of experts who voted for Mr. A is half of the total number of experts. Let's represent this as "x/2".

- Two-thirds voted for Mr. B: This means that the number of experts who voted for Mr. B is two-thirds of the total number of experts. Let's represent this as "2x/3".

- 10 experts voted for both Mr. A and Mr. B: This means that there are 10 experts who voted for both Mr. A and Mr. B. Let's represent this as "10".

- 6 experts did not vote for either Mr. A or Mr. B: This means that there are 6 experts who did not vote for either Mr. A or Mr. B. Let's represent this as "6".

Now, let's draw a Venn diagram to represent this information:

```
A B
_________
| |
| |
| |
|_________|
```

From the information given, we know that the number of experts who voted for Mr. A is "x/2" and the number of experts who voted for Mr. B is "2x/3". We also know that there are 10 experts who voted for both Mr. A and Mr. B.

Let's fill in the Venn diagram with these values:

```
A B
_________
| |
| | 10
| |
|_________|
```

We also know that there are 6 experts who did not vote for either Mr. A or Mr. B. Let's add this information to the Venn diagram:

```
A B
_________
| |
| 6 | 10
| |
|_________|
```

Now, we need to find the value of "x". We can do this by adding up the number of experts in each region of the Venn diagram:

Number of experts who voted for Mr. A: x/2 + 10
Number of experts who voted for Mr. B: 2x/3 + 10
Number of experts who did not vote for either: 6

Since the total number of experts is the sum of these three values, we can set up the equation:

(x/2 + 10) + (2x/3 + 10) + 6 = x

Simplifying this equation, we get:

x/2 + 2x/3 + 26 = x

Multiplying through by 6 to eliminate the fractions, we get:

3x + 4x + 156 = 6x

7x + 156 = 6x

x = 156

Therefore, the total number of experts is 156.

But we need to find half of this number, because half the number of experts voted for Mr. A. So, the answer is 156/2 = 78.

Therefore, the correct answer is option C) 24.