Twenty eight matches were played in a football tournament. Each team met its opponent only once. The number of teams that took part in the tournament, is:

Options

A. a. 8
B. b. 7
C. c. 12
D. d. none of these

Answer

To solve the problem of finding the number of teams that participated in a football tournament where 28 matches were played, we follow these steps:

### Step 1: Understand the relationship between matches and teams
In a tournament where each team plays every other team exactly once, the total number of matches played can be represented using combinations. Specifically, if there are \( n \) teams, the number of matches played is given by the combination formula \( C(n, 2) \), which represents the number of ways to choose 2 teams from \( n \) teams.

### Step 2: Set up the equation
We know from the problem that there were 28 matches played. Therefore, we can set up the equation:
\[ C(n, 2) = 28 \]
This can be expressed mathematically as:
\[ \frac{n(n-1)}{2} = 28 \]

### Step 3: Solve for \( n \)
To eliminate the fraction, we multiply both sides of the equation by 2:
\[ n(n-1) = 56 \]
Next, we rearrange this into a standard quadratic equation:
\[ n^2 - n - 56 = 0 \]

### Step 4: Factor the quadratic equation
Now we need to factor the quadratic equation. We look for two numbers that multiply to -56 and add to -1. The numbers -8 and 7 fit this requirement:
\[ (n - 8)(n + 7) = 0 \]

### Step 5: Solve for \( n \)
Setting each factor to zero gives us:
1. \( n - 8 = 0 \) → \( n = 8 \)
2. \( n + 7 = 0 \) → \( n = -7 \) (we discard this solution since the number of teams cannot be negative)

Thus, the only valid solution is:
\[ n = 8 \]

### Conclusion
The number of teams that took part in the tournament is \( 8 \).

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