A handshake occurs uniquely between two individuals. For $n$ people, the total number of unique handshakes corresponds to the combination of $n$ items taken 2 at a time:
$\text{Total handshakes} = \binom{n}{2} = \frac{n(n - 1)}{2}$
For $n = 20$:
$\text{Handshakes} = \frac{20 \times 19}{2} = 10 \times 19 = 190$.

Option analysis:

• 180: Calculated as $20 \times 9$.
• 155: Numerical error.
• 145: Numerical error.
• 190: Correct total number of handshakes.