Solving with the Principle of Inclusion-Exclusion

This problem can be solved using an algebraic approach based on the formula for the union of two sets, also known as the Principle of Inclusion-Exclusion.

1. Define the Variables

• Let T be the total number of experts.
• Let A be the number of experts who voted for Mr. A.
• Let B be the number of experts who voted for Mr. B.
• Let A ∩ B be the number of experts who voted for both.
• Let Neither be the number of experts who voted for neither.

2. Formulate the Equations

From the problem statement, we can write the following expressions:

• A = T / 2
• B = (2/3) × T
• A ∩ B = 10
• Neither = 6

The total number of experts is the sum of those who voted for at least one candidate plus those who voted for neither. The formula is:

T = A + B - (A ∩ B) + Neither

3. Solve for T

Now we substitute our expressions into the formula:

T = (T / 2) + ((2/3) × T) - 10 + 6

T = T/2 + 2T/3 - 4

To eliminate the fractions, we can multiply the entire equation by the common denominator, which is 6:

6T = 3T + 4T - 24

6T = 7T - 24

24 = 7T - 6T

T = 24

Conclusion

There were 24 experts in all.