To find the total number of handshakes among 20 students, we can start with the total possible handshakes if everyone shakes hands with everyone else.

1. Total possible handshakes: The formula for the total number of handshakes (combinations of 2 from $n$) is given by:

   $\text{Total handshakes} = \frac{n(n-1)}{2}$

   For $n = 20$:

   $\text{Total handshakes} = \frac{20 \times 19}{2} = 190$

2. Adjust for refusals:

   • Let $b$ be the number of boys and $g$ be the number of girls in the class such that $b + g = 20$.
   • Half of the boys refuse to shake hands with one third of the girls.

   Let's denote:

   • Half of the boys = $\frac{b}{2}$
   • One third of the girls = $\frac{g}{3}$

   The number of handshakes not made is:

   $\text{Refused handshakes} = \frac{b}{2} \times \frac{g}{3}$

   The total handshakes would then be:

   $\text{Actual handshakes} = 190 - \left(\frac{b}{2} \times \frac{g}{3}\right)$

3. Finding possible values: Since $b + g = 20$, we can explore different values of $b$ and $g$.

   • For example, if $b = 10$ and $g = 10$:

     $\text{Refused handshakes} = \frac{10}{2} \times \frac{10}{3} = 5 \times \frac{10}{3} = \frac{50}{3} \approx 16.67 \text{ (not an integer)}$

   • If $b = 12$ and $g = 8$:

     $\text{Refused handshakes} = \frac{12}{2} \times \frac{8}{3} = 6 \times \frac{8}{3} = 16 \text{ (valid)}$ $\text{Actual handshakes} = 190 - 16 = 174$

   • If $b = 14$ and $g = 6$:

     $\text{Refused handshakes} = \frac{14}{2} \times \frac{6}{3} = 7 \times 2 = 14 \text{ (valid)}$ $\text{Actual handshakes} = 190 - 14 = 176$

   • If $b = 16$ and $g = 4$:

     $\text{Refused handshakes} = \frac{16}{2} \times \frac{4}{3} = 8 \times \frac{4}{3} = \frac{32}{3} \approx 10.67 \text{ (not an integer)}$

   • If $b = 18$ and $g = 2$:

     $\text{Refused handshakes} = \frac{18}{2} \times \frac{2}{3} = 9 \times \frac{2}{3} = 6 \text{ (valid)}$ $\text{Actual handshakes} = 190 - 6 = 184$

From these calculations, the valid options for total handshakes are 190 and 184.

Thus, the possible total number of handshakes from the given options is 184.

Would you like to explore further details or have any questions? Here are some follow-up questions:

1. What is the total number of handshakes if all students shake hands?
2. How would the result change if all students shook hands without any refusals?
3. Can you explain why the refusal to shake hands affects the total?
4. How would you calculate handshakes in a different sized class?
5. What if instead, all boys and girls shook hands with only their respective genders?

Tip: Always check the calculations for refusals carefully, as they can affect the final tally significantly.