In a football championship 153 matches were played. Every team played one match with each other team. How many teams participated in the championship?

Options

A. 21
B. 18
C. 17
D. 15

Answer

To solve the problem of how many teams participated in the football championship where 153 matches were played, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
Each team plays one match against every other team. If there are \( n \) teams, the total number of matches played can be determined using the formula for combinations, specifically \( C(n, 2) \), which represents the number of ways to choose 2 teams from \( n \) teams to play a match.

2. **Using the Combination Formula**:
The formula for combinations is given by:
\[ C(n, 2) = \frac{n(n-1)}{2} \]
Since we know that 153 matches were played, we can set up the equation:
\[ \frac{n(n-1)}{2} = 153 \]

3. **Eliminating the Fraction**:
To eliminate the fraction, multiply both sides of the equation by 2:
\[ n(n-1) = 306 \]

4. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ n^2 - n - 306 = 0 \]

5. **Applying the Quadratic Formula**:
We can solve this quadratic equation using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -1 \), and \( c = -306 \). Plugging in these values:
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-306)}}{2 \cdot 1} \]
\[ n = \frac{1 \pm \sqrt{1 + 1224}}{2} \]
\[ n = \frac{1 \pm \sqrt{1225}}{2} \]
\[ n = \frac{1 \pm 35}{2} \]

6. **Calculating the Possible Values of \( n \)**:
This gives us two potential solutions:
\[ n = \frac{36}{2} = 18 \quad \text{and} \quad n = \frac{-34}{2} = -17 \]

7. **Selecting the Valid Solution**:
Since \( n \) must be a positive integer (you cannot have a negative number of teams), we discard \( n = -17 \). Thus, we have:
\[ n = 18 \]

8. **Conclusion**:
Therefore, the number of teams that participated in the championship is \( \boxed{18} \).