Six players are to play doubles Tennis match. IF every possible player must play with every other possible player, then the number of matches to be played is

Options

A. 30
B. 45
C. 60
D. 90

Answer

To solve the problem of how many doubles tennis matches can be played with 6 players, where every possible player must play with every other possible player, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- In a doubles tennis match, there are 4 players involved: 2 players on one team and 2 players on the other team.
- We need to find out how many unique matches can be formed with 6 players.

2. **Selecting Players**:
- First, we need to select 4 players from the 6 players available. The number of ways to choose 4 players from 6 can be calculated using the combination formula \( \binom{n}{r} \), where \( n \) is the total number of players and \( r \) is the number of players to choose.
- Here, \( n = 6 \) and \( r = 4 \).
- The formula is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

3. **Calculating Combinations**:
- We calculate \( \binom{6}{4} \):
\[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \cdot 2!} \]
- Simplifying this:
\[ = \frac{6 \times 5}{2 \times 1} = 15 \]
- So, there are 15 ways to select 4 players from 6.

4. **Forming Teams**:
- Once we have selected 4 players, we need to form 2 teams of 2 players each. The number of ways to divide 4 players into 2 teams of 2 can be calculated as follows:
- The number of ways to choose 2 players from 4 is \( \binom{4}{2} \), but since the order of teams does not matter (i.e., Team A vs Team B is the same as Team B vs Team A), we need to divide by 2:
\[ \text{Ways to form teams} = \frac{\binom{4}{2}}{2} = \frac{6}{2} = 3 \]

5. **Total Matches**:
- Now, we multiply the number of ways to select the players by the number of ways to form teams:
\[ \text{Total Matches} = \binom{6}{4} \times 3 = 15 \times 3 = 45 \]

### Final Answer:
The total number of matches that can be played is **45**.

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