Question

In a class tournament, each of eight players will play exactly once. How many matches will be played during the tournament?

• A. 28
• B. 15
• C. 56
• D. 30

Hint:

The number of matches in a round-robin tournament is given by \( \binom{n}{2} \), where \( n \) is the number of players.

Answer 1

The Correct Option is A

To determine the number of matches played during the tournament where each of the eight players plays against every other player exactly once, we can use a combination formula. Here, the task is to choose 2 players from a group of 8 to form a match. This is a combination problem that can be represented as "8 choose 2", denoted mathematically as \( \binom{8}{2} \).
The formula for combinations is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] For our problem,
- \( n = 8 \) (total players), - \( r = 2 \) (players per match).
Substituting the values, we have:
\[ \binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8!}{2! \cdot 6!} \]
Calculating factorials:
- \( 8! = 8 \times 7 \times 6! \) - \( 2! = 2 \) - \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \) Substitute and simplify:
\[ \binom{8}{2} = \frac{8 \times 7 \times 6!}{2 \times 6!} = \frac{8 \times 7}{2} = \frac{56}{2} = 28 \]
Thus, a total of 28 matches will be played during the tournament.