Understanding the Class Friend Relationships

The problem describes a class of 30 students and involves the friendships of two students, L and A. We are given information about the number of friends L has, the number of friends A has, and the number of friends they have in common. We need to find out how many students in the class are friends with neither L nor A.

This type of problem can be effectively solved using the principles of set theory or by visualizing it with a Venn diagram. We can consider the set of L's friends and the set of A's friends within the universal set of all students in the class.

Analyzing the Given Information

Let's break down the information provided:

• Total number of students in the class = 30
• L has 10 unique friends (friends L has, but A does not).
• L and A have 5 friends who are common to both (friends they share).
• A has a total of 17 friends in the class.

Calculating Friend Sets

From the given information, we can calculate the size of different friend sets:

1. L's total friends: The total number of friends L has is the sum of L's unique friends and their common friends with A.
   Total friends of L = L's unique friends + Common friends
   Total friends of L = $10 + 5 = 15$
2. A's unique friends: A has a total of 17 friends, and 5 of these are also friends with L (the common friends). So, A's unique friends are A's total friends minus the common friends.
   A's unique friends = A's total friends - Common friends
   A's unique friends = $17 - 5 = 12$

Finding Students Who Are Friends with L or A or Both

To find the number of students who are friends with at least one of them (either L or A or both), we can sum the unique friends of L, the unique friends of A, and the friends common to both.

Number of students who are friends with L or A or both = L's unique friends + A's unique friends + Common friends

Number of students who are friends with L or A or both = $10 + 12 + 5 = 27$

Alternatively, using the Principle of Inclusion-Exclusion:

$|L \cup A| = |L| + |A| - |L \cap A|$

Where:

• $|L|$ is the total number of L's friends (which we calculated as 15).
• $|A|$ is the total number of A's friends (given as 17).
• $|L \cap A|$ is the number of common friends (given as 5).

So, $|L \cup A| = 15 + 17 - 5 = 32 - 5 = 27$.

This confirms that 27 students are friends with L or A or both.

Calculating Students Friends with Neither L Nor A

The total number of students in the class is 30. We found that 27 students are friends with at least L or A. The students who are friends with neither L nor A are those students who are not included in the group of 27.

Number of students friends with neither = Total students in the class - Number of students friends with L or A or both

Number of students friends with neither = $30 - 27 = 3$

Therefore, 3 students are friends with neither L nor A.

Conclusion

Based on the calculations, in a class of 30 students, there are 3 students who are friends with neither L nor A.

Revision Table - Friend Relationships in Class

Review of the key numbers and calculations:

• Total students: 30
• L's unique friends: 10
• Common friends (L and A): 5
• A's total friends: 17
• L's total friends = $10 + 5 = 15$
• A's unique friends = $17 - 5 = 12$
• Friends of L or A or both = L's unique friends + A's unique friends + Common = $10 + 12 + 5 = 27$
• Friends with neither = Total students - Friends of L or A or both = $30 - 27 = 3$

Additional Information - Set Theory and Venn Diagrams

This problem is a practical application of basic set theory concepts, often visualized using Venn diagrams. Let $S$ be the set of all students, $F_L$ be the set of friends of L, and $F_A$ be the set of friends of A.

• The total number of students is $|S| = 30$.
• L's unique friends represent the set difference $F_L \setminus F_A$, with $|F_L \setminus F_A| = 10$.
• Common friends represent the intersection $F_L \cap F_A$, with $|F_L \cap F_A| = 5$.
• A's total friends represent $|F_A| = 17$.

We calculated $|F_L| = |F_L \setminus F_A| + |F_L \cap F_A| = 10 + 5 = 15$.

We calculated A's unique friends, which is the set difference $F_A \setminus F_L$, with $|F_A \setminus F_L| = |F_A| - |F_L \cap F_A| = 17 - 5 = 12$.

The set of students who are friends with L or A or both is the union $F_L \cup F_A$. Its size is given by the formula:

$|F_L \cup F_A| = |F_L \setminus F_A| + |F_A \setminus F_L| + |F_L \cap F_A|$

$|F_L \cup F_A| = 10 + 12 + 5 = 27$

Alternatively, using the Principle of Inclusion-Exclusion:

$|F_L \cup F_A| = |F_L| + |F_A| - |F_L \cap F_A|$

$|F_L \cup F_A| = 15 + 17 - 5 = 27$

The students who are friends with neither L nor A are those outside the union $F_L \cup F_A$ within the universal set $S$. This is the complement of the union, $(F_L \cup F_A)^c$.

$|(F_L \cup F_A)^c| = |S| - |F_L \cup F_A|$

$|(F_L \cup F_A)^c| = 30 - 27 = 3$

This demonstrates how set theory confirms the step-by-step calculation.