what is the equation for this problem?

Question

There are six teams in a basket ball game. Each team will play each other once . How many total games will be played ? What if there was 20 teams? What if there were 12 teams? Can you find a rule that will let you predict the number of games for any number of teams? Write equation in function form. (table to).

Answer 1

If there are 6 teams then each team has 5 opponents to play against, so it might seem at first glance that 30 games would be played.  But each match involves 2 teams so we have double counted A versus B and B versus A.  So the total number of games played will be 6*5/2 = 15.  

 

You can try working the problem for smaller numbers of teams to come up with a general rule.  For example,  

 

if there are 2 teams there will be 2*1/2 = 1 game  played

if there are 3 teams there will be 3*2/2 = 3 games played  

if there are 4 teams there will be 4*3/2 = 6 games played

if there are 5 teams there will be 5*4/2 = 10 games played  

 

Can you figure out the rule if there are N teams and use it to determine the number of games played if there were 12 or 20 teams?  

Answer 2

Hi Jacque,

 

If we pick any one of the teams from a group of six teams, how many other teams can that team play?

If we then pick a second team from a group of six teams, how many teams can that team play, not including the first team?

Similarly, if we pick a third team, how many teams can that team play, not including the first and second teams?

 

You'll notice a pattern as you go from the first team to the sixth team. The number of distinct games each team plays changes by how much?

 

 

We can apply this idea to 12 teams and 20 teams as well. 

If we pick any one of the teams in a group of twelve teams, how many other teams can that team play?
If we then pick a second team from a group of twelve teams, how many teams can that team play, not including the first team?

Similarly, if we pick a third team, how many teams can that team play, not including the first and second teams?

 

Whether there are six teams, twelve teams, or twenty teams, the second to last team we pick in each group will play one distinct game against the last team. What equation can we make based on these observations?