At the faculty costume party , each teacher from Transylvania Middle school came dressed as his of her favorite creepy creature . The teachers arrived one at a time , and when a new teacher came in , each of the teachers already there lined up at the door and shook hands with newcomer . Count Dracula (Mr.Blab, the math teacher) kept track of the # of handshakes exchanged . He said that there were 171handshakes exchanged at the party . How many teachers came to the costume party.

0,1,3,9,15,17,19 are my answer's

n(n-1)/2 = 171

Since 342 = 18*19, there were 19 teachers.

To determine the number of teachers who came to the costume party based on the given information, we can use a process of elimination.

Let's start by assuming there were 0 teachers at the party initially. In this case, there wouldn't be any handshakes, so the answer cannot be 0.

Next, we assume there was 1 teacher at the party initially. The newcomer would shake hands with the only teacher there, resulting in 1 handshake. This does not match the given total of 171 handshakes, so the answer cannot be 1.

We continue this process until we find a scenario that matches the given number of handshakes.

Assuming there were 3 teachers at the party initially:
- When the second teacher arrives, they shake hands with the first teacher, resulting in 1 handshake.
- When the third teacher arrives, they shake hands with the first and second teachers, resulting in 2 handshakes.

So, with 3 teachers, there would be a total of 3 handshakes.

This scenario does not match the given total of 171 handshakes, so the answer cannot be 3.

We must continue this process to find the correct number of teachers.

Assuming there were 9 teachers at the party initially:
- When the second teacher arrives, they shake hands with the 8 teachers already there, resulting in 8 handshakes.
- When the third teacher arrives, they shake hands with the 8 teachers already there, resulting in 8 handshakes.
- When the fourth teacher arrives, they shake hands with the 8 teachers already there, resulting in 8 handshakes.
- And so on, until the eighth teacher arrives, shaking hands with the 8 teachers already there, resulting in 8 handshakes.

In this scenario, each new teacher shakes hands with 8 teachers already there. Since there are 8 possible handshakes for each teacher, the number of handshakes is (number of teachers - 1) times 8.

For 9 teachers, the number of handshakes would be (9 - 1) * 8 = 8 * 8 = 64 handshakes.

This means that 9 teachers cannot account for the given total of 171 handshakes.

Continuing the process, we assume there were 15 teachers, but this also doesn't match the total of 171 handshakes.

Moving on to 17 teachers:
- When the second teacher arrives, they shake hands with the 16 teachers already there, resulting in 16 handshakes.
- When the third teacher arrives, they shake hands with the 16 teachers already there, resulting in 16 handshakes.
- And so on, until the 16th teacher arrives, shaking hands with the 16 teachers already there, resulting in 16 handshakes.

In this scenario, each new teacher shakes hands with 16 teachers already there. Since there are 16 possible handshakes for each teacher, the number of handshakes is (number of teachers - 1) times 16.

For 17 teachers, the number of handshakes would be (17 - 1) * 16 = 16 * 16 = 256 handshakes.

This means that 17 teachers cannot account for the given total of 171 handshakes.

The process continues with 19 teachers, but this also doesn't match the total of 171 handshakes.

Based on the elimination process, we can conclude that the answer is not 0, 1, 3, 9, 15, 17, or 19.

Thus, we need to consider other possibilities or scenarios to find the correct number of teachers who came to the costume party.