In a badminton tournament each player played one game with all the other players. How many players participated in the touinament if they played 105 games in all?

Options

A. 35
B. 12
C. 15
D. none the these

Answer

To solve the problem of how many players participated in a badminton tournament where 105 games were played, we can use the concept of combinations. Each player plays one game with every other player, which means we can represent the total number of games played as the combination of players taken 2 at a time.

### Step-by-Step Solution:

1. **Understand the Formula**: The total number of games played in a tournament where each player plays with every other player is given by the formula:
\[ \text{Total Games} = \binom{n}{2} = \frac{n(n-1)}{2} \]
where \( n \) is the number of players.

2. **Set Up the Equation**: Given that the total number of games played is 105, we can set up the equation:
\[ \frac{n(n-1)}{2} = 105 \]

3. **Multiply Both Sides by 2**: To eliminate the fraction, multiply both sides of the equation by 2:
\[ n(n-1) = 210 \]

4. **Rearrange the Equation**: Rearranging gives us a quadratic equation:
\[ n^2 - n - 210 = 0 \]

5. **Factor the Quadratic Equation**: We need to factor the quadratic equation. We are looking for two numbers that multiply to -210 and add to -1. The numbers are 14 and -15. Thus, we can write:
\[ (n - 15)(n + 14) = 0 \]

6. **Solve for \( n \)**: Setting each factor equal to zero gives us:
\[ n - 15 = 0 \quad \Rightarrow \quad n = 15 \]
\[ n + 14 = 0 \quad \Rightarrow \quad n = -14 \quad (\text{not valid since } n \text{ must be positive}) \]

7. **Conclusion**: The number of players \( n \) is 15.

### Final Answer:
Thus, the total number of players who participated in the tournament is **15**.

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