In a class tournament where the participants were to play one game with one another, two of the class players fell ill, having played 3 games each. If the total number of games played is 24, then the number of participants at the beginning was

Answer

To solve the problem, we need to determine the number of participants in the class tournament based on the given conditions. Let's break down the solution step by step.

### Step 1: Understand the problem
We know that there were initially \( n \) participants in the tournament. Two players fell ill after playing 3 games each. The total number of games played in the tournament is 24.

### Step 2: Calculate the games played by the two ill players
Since the two players played 3 games each, the total number of games played by these two players is:
\[ 3 + 3 = 6 \text{ games} \]

### Step 3: Calculate the games played by the remaining participants
Since the total number of games played is 24, the games played by the remaining participants (those who did not fall ill) is:
\[ 24 - 6 = 18 \text{ games} \]

### Step 4: Set up the equation for remaining participants
Let the number of remaining participants be \( n - 2 \) (since 2 players fell ill). The number of games played among \( n - 2 \) participants can be calculated using the combination formula for choosing 2 players from \( n - 2 \):
\[ \text{Games played} = \binom{n-2}{2} = \frac{(n-2)(n-3)}{2} \]

### Step 5: Set up the equation
We know that the number of games played among the remaining participants is 18, so we can set up the equation:
\[ \frac{(n-2)(n-3)}{2} = 18 \]

### Step 6: Solve the equation
Multiply both sides by 2 to eliminate the fraction:
\[ (n-2)(n-3) = 36 \]

Expanding the left side gives:
\[ n^2 - 5n + 6 = 36 \]

Rearranging the equation:
\[ n^2 - 5n + 6 - 36 = 0 \]
\[ n^2 - 5n - 30 = 0 \]

### Step 7: Factor the quadratic equation
Now, we will factor the quadratic equation:
\[ (n - 10)(n + 3) = 0 \]

### Step 8: Find the possible values for \( n \)
Setting each factor to zero gives us:
1. \( n - 10 = 0 \) → \( n = 10 \)
2. \( n + 3 = 0 \) → \( n = -3 \) (not a valid solution since the number of participants cannot be negative)

Thus, the only valid solution is:
\[ n = 10 \]

### Conclusion
The number of participants at the beginning was **10**.

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