25 persons are in a room. 15 of them play hockey, 17 of them play football and 10 of them play both hockey and football. Then the number of persons playing neither hockey nor football is:
3
This question involves applying basic principles of set theory to determine the number of individuals who do not participate in either of two given activities. We are provided with the total number of persons in a room, the number of persons playing hockey, the number of persons playing football, and the number of persons playing both sports. Our goal is to find the number of persons playing neither hockey nor football.
Persons in the Room: Understanding the Data
Let's first list down the information provided in the question related to the persons playing sports:
- Total number of persons in the room = 25
- Number of persons who play hockey = 15
- Number of persons who play football = 17
- Number of persons who play both hockey and football = 10
Key Concepts for Persons Playing Sports
To solve this problem, we will use the principles of set theory. Let H be the set of persons who play hockey and F be the set of persons who play football. The total number of persons is represented by the universal set U.
The formula for the union of two sets, which represents the number of persons playing at least one of the two sports (hockey or football), is given by:
\(N(H \cup F) = N(H) + N(F) - N(H \cap F)\)
Where:
- \(N(H)\) is the number of persons playing hockey.
- \(N(F)\) is the number of persons playing football.
- \(N(H \cap F)\) is the number of persons playing both hockey and football (the intersection).
- \(N(H \cup F)\) is the number of persons playing at least one of the sports (the union).
Once we find the number of persons playing at least one sport, we can determine the number of persons playing neither sport by subtracting this value from the total number of persons in the room.
\(N(\text{neither H nor F}) = N(U) - N(H \cup F)\)
Calculating Persons Playing Neither Sport
Let's perform the calculations step-by-step using the given data about the persons.
Step 1: Calculate the number of persons playing at least one sport (Hockey or Football)
Using the formula for the union of sets:
\(N(H \cup F) = N(H) + N(F) - N(H \cap F)\)
Substitute the given values into the formula:
\(N(H \cup F) = 15 + 17 - 10\)
\(N(H \cup F) = 32 - 10\)
\(N(H \cup F) = 22\)
So, 22 persons play at least one of the two sports (hockey or football).
Step 2: Calculate the number of persons playing neither hockey nor football
The total number of persons in the room is 25. To find the number of persons who play neither sport, we subtract the number of persons playing at least one sport from the total number of persons:
\(N(\text{neither H nor F}) = N(U) - N(H \cup F)\)
\(N(\text{neither H nor F}) = 25 - 22\)
\(N(\text{neither H nor F}) = 3\)
Therefore, 3 persons play neither hockey nor football.
Summary of Persons Playing Statistics
Here is a summary of the data and the calculated result, helping to visualize the breakdown of persons:
| Category | Number of Persons |
|---|---|
| Total Persons in the Room | 25 |
| Persons Playing Hockey Only | \(15 - 10 = 5\) |
| Persons Playing Football Only | \(17 - 10 = 7\) |
| Persons Playing Both Hockey and Football | 10 |
| Persons Playing At Least One Sport | 22 |
| Persons Playing Neither Hockey Nor Football | 3 |
The number of persons playing neither hockey nor football is 3.
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