To find the number of handshakes, we can use the combination formula.
In this case, there are 12 members at the meeting, and each member shakes hands with all the other members.
To calculate the number of handshakes, we need to find the number of combinations of 2 members chosen from a group of 12.
The formula for combinations is given by:
$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$
In this case, we have:
$$\binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12!}{2!10!}$$
Calculating the factorials:
$$12! = 12 \times 11 \times 10!$$
$$2! = 2 \times 1$$
$$10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$$
Plugging in the values:
$$\binom{12}{2} = \frac{12 \times 11 \times 10!}{2 \times 1 \times 10!}$$
Simplifying:
$$\binom{12}{2} = \frac{12 \times 11}{2}$$
$$\binom{12}{2} = 66$$
Therefore, the correct answer is option C) 66.