In a badminton each player played one game with ll the other players. Number of players participated in the tournament if they played 105 games is

Options

A. 35
B. 15
C. 12
D. 10

Answer

To solve the problem of determining the number of players who participated in a badminton tournament where each player played one game with every other player, and a total of 105 games were played, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
Each player plays against every other player exactly once. If there are \( n \) players, the total number of games played can be calculated using the combination formula for choosing 2 players from \( n \), which is given by:
\[ \text{Total Games} = \binom{n}{2} = \frac{n(n-1)}{2} \]

2. **Setting Up the Equation**:
We know from the problem statement that the total number of games played is 105. Therefore, we can set up the equation:
\[ \frac{n(n-1)}{2} = 105 \]

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

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

5. **Factoring 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 -15 and 14 work:
\[ (n - 15)(n + 14) = 0 \]

6. **Finding the Roots**:
Setting each factor to zero gives us:
\[ n - 15 = 0 \quad \text{or} \quad n + 14 = 0 \]
This results in:
\[ n = 15 \quad \text{or} \quad n = -14 \]

7. **Interpreting the Results**:
Since the number of players cannot be negative, we discard \( n = -14 \). Thus, the number of players is:
\[ n = 15 \]

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

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