T.J. is hosting a dinner party. T.J. greets the first guest and they shake hands. The second guest arrives and shakes hands with T.J. and the first guest. The third guest arrives and shakes hands with T.J., the first guest, and the second guest. This pattern continues. How many handshakes have taken place after the ninth guest arrives?

Let's calculate the number of handshakes using the pattern described.

- When the first guest arrives, there's only one handshake (with T.J.). - When the second guest arrives, they shake hands with T.J. and the first guest, which is 2 handshakes. - The third guest shakes hands with T.J., the first guest, and the second guest, resulting in 3 more handshakes.

This pattern shows that each new guest shakes hands with all the people who have already arrived, including T.J.

So, the total number of handshakes after the ninth guest arrives can be calculated by adding up the handshakes for each guest:

1 (with the first guest) + 2 (with the second guest) + 3 (with the third guest) + 4 (with the fourth guest) + 5 (with the fifth guest) + 6 (with the sixth guest) + 7 (with the seventh guest) + 8 (with the eighth guest) + 9 (with the ninth guest) = 1+2+3+4+5+6+7+8+9 = 45 handshakes in total, after the ninth guest has arrived.