The image shows an assignment from MAT 203: Discrete Mathematical Structures. The assignment contains four questions related to lattices, Hasse diagrams, binomial expansions, and recurrence relations. Here's a breakdown of the questions:

1. Define distributive lattice. Show that $(\mathbb{Z}, \leq)$ is a distributive lattice.
   Marks: 1

2. Draw the Hasse diagram of $D_{42}$. Find the complements of each of its elements.
   Marks: 2

3. Find the coefficient of $x^{20}$ in the expansion of $f(x) = (x^2 + x^3 + x^4 + x^5 + x^6)^5$.
   Marks: 1

4. Solve the recurrence relation: $a_{n+2} - 8a_{n+1} + 19a_n = 8(5^n) + 6(4^n), \quad n \geq 2$ Given $a_0 = 12$ and $a_1 = 5$.
   Marks: 2

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Here are 5 related questions that expand on this material:

1. How can you prove that every distributive lattice is also a modular lattice?
2. What is the general method to find complements in any lattice?
3. How does one construct a Hasse diagram for any given poset?
4. How do you generalize the binomial theorem for multiple terms?
5. How can recurrence relations with non-homogeneous terms be solved using the method of undetermined coefficients?

Tip: Always start by understanding the basic properties of algebraic structures like lattices before applying more complex results like distributivity.