In a class of 45 students, each student likes to play either badminton or hockey. There are 20 students who like to play badminton, and 5 students like to play both badminton and hockey. Then, the number of students who play hockey only is

We can solve this problem using the principle of set theory. Let's define the following:

- Let \( B \) be the set of students who like to play badminton.
- Let \( H \) be the set of students who like to play hockey.

We are given:
- Total number of students = 45
- Number of students who like badminton, \( |B| = 20 \)
- Number of students who like both badminton and hockey, \( |B \cap H| = 5 \)

We need to find the number of students who play **only hockey**.

### Step 1: Find the number of students who play hockey.

Let \( x \) be the number of students who like only hockey. The total number of students who like hockey (including those who like both hockey and badminton) is represented by \( |H| \).

Now, using the inclusion-exclusion principle for the total number of students:

\[
|B \cup H| = |B| + |H| - |B \cap H|
\]

The total number of students is given as 45, which means:

\[
45 = 20 + |H| - 5
\]

Simplifying:

\[
45 = 15 + |H|
\]

\[
|H| = 30
\]

### Step 2: Find the number of students who play only hockey.

The number of students who like only hockey is:

\[
\text{Number of students who play only hockey} = |H| - |B \cap H| = 30 - 5 = 25
\]

Thus, the number of students who play only hockey is **25**.

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Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. How many students like only badminton?
2. How many students like only one sport?
3. How many students like at least one of the two sports?
4. If 10 more students join who only like hockey, how many students now like hockey?
5. What percentage of students like both badminton and hockey?

**Tip:** In set theory problems, the inclusion-exclusion principle is often useful to avoid double-counting elements common to two or more sets.

In this math problem, learn how to apply set theory and the inclusion-exclusion principle to find the number of students who play only hockey in a class of 45. Given that 20 students like badminton and 5 like both sports, discover how to solve the problem step by step. Suitable for students in Grades 8-10.

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