Everybody in $a$ room shakes hands with everybody else. The total number of shake hands is 66. The number of persons in the room is ________.

Step 1 (Handshake Formula): If there are $n$ people in a room and each person shakes hands with everyone else exactly once, the total number of handshakes is given by choosing a pair of people out of $n$:

Step 2 (Set up the Equation): We are given the total handshakes is 66:

Step 3 (Solve the Quadratic Equation):

This gives two solutions $n=12$ and $n=-11$. Since the number of people $n$ must be positive, we choose $n=12$.

Step 4 (Conclusion): The number of persons in the room is exactly 12.