A soccer team just became first in the national league. The players which were on the field just after the referee blew the final signal wanted to congratulate each other by shaking hands. Assuming that each player shook hands with every other player only once, how many time were the hands shaken in total? (There are of course 11 players in a soccer team)

Just name the players 1, 2, 3, .... and 11. Player 1 has to shake hands with player 2, 3, ... and 11. So this counts for 10 shakes.
Player 2 has already shaken hands with player 1. He still has to shake hands with player 3, 4, ... and 11. So this counts for 9 shakes.
So continuing, player 3 has now to shake still 8 times, then player 4 7 times, etc. So in total hands were shaken 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55 times.