Among 160 players in a tournament, 57 did not participate in any of the three games, i.e. Cricket, Hockey and Badminton. A total of 37 players participated in only one game, 10 players participated in both Cricket and Hockey but not in Badminton, 9 players participated in both Hockey and Badminton but not in Cricket, and 13 players participated in both Cricket and Badminton but not in Hockey. How many players participated in all the three games?
34
Tournament Player Participation Analysis
This problem requires us to determine the number of players involved in all three games (Cricket, Hockey, Badminton) within a tournament. We are given the total number of players, the count of those who sat out entirely, and details about participation in single games and specific overlaps between two games. We need to use this information to find the number of players participating in all three sports.
Key Tournament Participation Data
Here's a summary of the information provided:
- Total players in the tournament: 160
- Players not participating in any game (Cricket, Hockey, Badminton): 57
- Players participating in exactly one game: 37
- Players participating in Cricket and Hockey only: 10
- Players participating in Hockey and Badminton only: 9
- Players participating in Cricket and Badminton only: 13
Calculating Active Participants
First, let's find the number of players who participated in at least one game. This is the total number of players minus those who didn't participate in any game.
Number of players in at least one game = Total players - Players not in any game
Number of players in at least one game = $160 - 57 = 103$
So, $103$ players actively participated in the tournament across the three games.
Participation Categories Breakdown
The total number of active participants ($103$) can be broken down into three mutually exclusive groups:
- Those who played exactly one game.
- Those who played exactly two games.
- Those who played exactly three games.
We can write this relationship as:
Players in at least one game = (Players in exactly one game) + (Players in exactly two games) + (Players in exactly three games)
Calculating Players in Exactly Two Games
We need to sum the counts of players who participated in specific pairs of games but not the third:
- Cricket and Hockey only: 10
- Hockey and Badminton only: 9
- Cricket and Badminton only: 13
Total players in exactly two games = $10 + 9 + 13 = 32$
Determining Players in All Three Games
Let '$x$' be the number of players who participated in all three games (Cricket, Hockey, and Badminton).
Now we can plug the known values into our participation breakdown equation:
Players in at least one game = (Players in exactly one game) + (Players in exactly two games) + (Players in exactly three games)
$103 = 37 + 32 + x$
Combine the numbers for one and two games:
$103 = 69 + x$
To find '$x$', subtract 69 from 103:
$x = 103 - 69$
$x = 34$
Final Answer
Based on the calculations, the number of players who participated in all three games (Cricket, Hockey, and Badminton) is 34.
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