Question

Several teams take part in a competition, each of which must play one game with all other teams. How many teams took part in the competition if they played forty five games in all?

Answer 1

Hint: The given question is a condition for total number of played games, in which we have to find the number of teams participated in the game, for such question you need to assume a variable for the solution and then derive the expression as per the condition given in the question, with that expression we can solve for the solution variable.

Complete step by step solution:
For the given question let us assume a variable say “x” which shows the total number of teams played a game, now the condition says that each team played one game with every other team, deriving this condition into mathematical form we can say:
\[
   \Rightarrow \text{let total number of teams} = x \\
 \]
Here the number of games is calculated by the given condition that each team will play a single match with all other teams.
Now we will choose two teams to complete a match so we will use the combination formula.
\[
   \Rightarrow ^xC_2= 45 \\
   \Rightarrow \dfrac{ x(x-1)}{2} = 45 \\
   \Rightarrow x(x-1) = 90 \\
 \]
After solving this equation we will get x=10
Hence the total number of teams played is 10.

Note: In the above question there is no condition for the rules of knockout in the match, hence we calculated for the single game played by all the teams, if there was given any condition for the match result, then we have to first calculate the winner and accordingly we can tell for the total number of teams.