The Southeastern Conference (SEC), has fourteen universities, each of which fields a football team. The football teams are divided into two seven-team divisions. Each season, every team plays every other team in its division once, and also plays two teams from the other division. Additionally, the two division champions meet in a championship game. How many football games every season involve two teams from the SEC?
a) Each team plays every other team in the division
Number of games = 7C2 = 21
b) Each team plays two teams from the other division
Total number of games = 14 (When one team plays another, both of them have one out of the two needed games completed. Seven teams play against seven teams (7) and then those seven teams play against seven different teams (7 more) )
c) Final game = 1 game
Total = 21 + 14 + 1 = 36
To find out how many football games involve two teams from the SEC, we need to calculate the number of games within each division and the number of games between divisions.
In each division, there are seven teams. To find the number of games within each division, we can use the formula for combinations. A combination is a way to select items from a larger set without regard to the order of the items. The formula for calculating combinations is:
nCr = n! / (r!(n-r)!)
Where n is the total number of items and r is the number of items being chosen. In this case, we need to choose 2 teams from each division, so we can calculate:
nCr = 7! / (2!(7-2)!) = 7! / (2!5!) = 7 * 6 / (2 * 1) = 21
So within each division, there are a total of 21 games.
Now, let's determine the number of games between divisions. Each team plays two teams from the other division. Since there are 7 teams in each division, there will be a total of 7 * 2 = 14 interdivisional games.
Finally, we need to consider the championship game, which involves the two division champions. Since we have two divisions, there will be 1 championship game.
To find the total number of games, we add the games within divisions, the interdivisional games, and the championship game:
Total games = Games within divisions + Interdivisional games + Championship game
Total games = 21 + 14 + 1 = 36
Therefore, there are 36 football games every season involving two teams from the SEC.