To solve the problem, we can use the principle of inclusion-exclusion and some basic set theory. Let's break down the information provided.
Given Data
- Total viewers: 500
- Football viewers (F): 285
- Hockey viewers (H): 195
- Basketball viewers (B): 115
- Viewers not watching any game: 50
- Football and Basketball viewers: 45
- Football and Hockey viewers: 70
- Hockey and Basketball viewers: 50
Calculating Viewers of All Three Games
Let \( x \) be the number of viewers who watch all three games. We can set up the following equation based on the inclusion-exclusion principle:
Using the formula:
F + H + B - (F \cap H) - (H \cap B) - (B \cap F) + x = Total Viewers
Substituting the values:
285 + 195 + 115 - 70 - 50 - 45 + x = 450
Calculating this gives:
285 + 195 + 115 - 70 - 50 - 45 = 410
So, we have:
410 + x = 450
Thus, \( x = 450 - 410 = 40 \).
Number of Viewers Watching All Three Games
40 viewers watch all three games.
Finding Viewers Who Watch Exactly One Game
To find the number of viewers who watch exactly one game, we can use the following formula:
Exactly one game = (Total viewers of game) - (Viewers of two games) - (Viewers of all three games).
Calculating Each Game
- Football only: \( 285 - (70 + 45 - 40) = 285 - 75 = 210 \)
- Hockey only: \( 195 - (70 + 50 - 40) = 195 - 80 = 115 \)
- Basketball only: \( 115 - (45 + 50 - 40) = 115 - 55 = 60 \)
Total Viewers Watching Exactly One Game
Now, adding these values together:
210 (Football only) + 115 (Hockey only) + 60 (Basketball only) = 385.
385 viewers watch exactly one of the three games.
