The image contains several mathematical exercises, primarily focused on set theory and cardinality. I will transcribe and explain each exercise for clarity:

Exercice 7:

Let $E$ be a set with $n$ elements, and let $A \subset E$ be a subset with $p$ elements. What is the number of subsets of $E$ that contain exactly one element from $A$?

Exercice 8:

Let $E$ be a finite set with cardinality $n$, and $A \subset E$ a subset of $E$. Determine the number of disjoint subsets of $E$ that are also subsets of $A$.

Exercice 9:

Let $E$ be a finite set with cardinality $n$, determine the cardinality of the set: $P(E) \cap \{ X \subset Y \}$

Exercice 10:

Let $E$ be a finite set with cardinality $n$, determine the cardinality of the set: $P(E)^2 : X \cap Y = \emptyset$ This refers to pairs of subsets of $E$ whose intersection is empty.

Exercice 11:

Let $E$ be a finite set with cardinality $n \geq 1$. Determine the number of subsets of $E$ with even cardinality.

Do you want detailed solutions for any of these exercises? Here are some follow-up questions that may help expand on this:

1. What are the general rules for counting subsets of a set with specific properties (e.g., disjoint subsets)?
2. How can the binomial coefficient $\binom{n}{k}$ be used to solve Exercise 7?
3. What is the significance of even cardinality in Exercise 11?
4. How does the concept of the power set $P(E)$ apply to Exercises 9 and 10?
5. How are disjoint sets defined and why is their intersection empty?

Tip: In problems involving subsets, particularly disjoint subsets or subsets with specific properties, understanding the structure of the power set is crucial.