Question

There are 8 teams in a certain league and each team plays each of the other teams exactly once. If each game is played by 2 teams, what is the total number of games played?
A) 15
B) 6
C) 28
D) 56
E) 64

Answer 1

Hint:
First, we will use the formula of combinations \[{}^n{C_r} = \dfrac{{\left. {\underline {\,
 n \,}}\! \right| }}{{\left. {\underline {\,
 r \,}}\! \right| \cdot \left. {\underline {\,
 {n - r} \,}}\! \right| }}\]. Then take \[n = 8\] and \[r = 2\] to find the total number of games played by 8 teams.

Complete step by step solution:
Given that there are 8 teams in a league.

We know that none of the team in a league can play against itself, so this question is an example of combinations.

We will find the total number of games played in a league using the combinations
\[{}^n{C_r} = \dfrac{{\left. {\underline {\,
 n \,}}\! \right| }}{{\left. {\underline {\,
 r \,}}\! \right| \cdot \left. {\underline {\,
 {n - r} \,}}\! \right| }}\], where \[n\] is the number of items, and \[r\] represents the number of items being chosen.

Here, there are 8 teams and each game is played by 2 teams, so we have
\[n = 8\]
\[r = 2\]

Substituting these values of \[n\] and \[r\] in \[{}^n{C_r} = \dfrac{{\left. {\underline {\,
 n \,}}\! \right| }}{{\left. {\underline {\,
 r \,}}\! \right| \cdot \left. {\underline {\,
 {n - r} \,}}\! \right| }}\], we get

\[
  {}^8{C_2} = \dfrac{{\left. {\underline {\,
 8 \,}}\! \right| }}{{\left. {\underline {\,
 2 \,}}\! \right| \cdot \left. {\underline {\,
 {8 - 2} \,}}\! \right| }} \\
   = \dfrac{{\left. {\underline {\,
 8 \,}}\! \right| }}{{\left. {\underline {\,
 2 \,}}\! \right| \cdot \left. {\underline {\,
 6 \,}}\! \right| }} \\
   = \dfrac{{8 \cdot 7 \cdot \left. {\underline {\,
 6 \,}}\! \right| }}{{\left. {\underline {\,
 2 \,}}\! \right| \cdot \left. {\underline {\,
 {8 - 2} \,}}\! \right| }} \\
   = \dfrac{{8 \cdot 7}}{2} \\
   = 28 \\
\]

Therefore, the total number of games played is 28.
Hence, the option C is correct.

Note:
We can also solve this question by picking any one team, naming it as team 1 and the rest as 2 to 8. Since none of the team can play the game with itself, so team 1 will be playing only seven games. Now, we will come to team 2 and as team 2 has already played a game with team 1, so only six games will be left. Thus, continuing in the same manner, we can find the total number of matches by adding the matches played by each team.
\[7 + 6 + 5 + 4 + 3 + 2 + 1 = 28\]
Hence, the total number of games played is 28.