---
title: Endterm Sample Spring 2026
tags:
  - Sample
  - Endterm
  - DiscreteMathematics
  - Maths
description: Discrete Mathematics - Endterm Sample - Spring 2026
---
# Endterm Sample Spring 2026
# Discrete Mathematics - Endterm Sample - Spring 2026

**NAME:**  
**NEPTUN code:**  
**Group number/Class Teacher:**

> [!abstract] Exam details
> 
> - **Duration:** 90 minutes.
> - **Total score:** 40 points.
> - **Success criterion:** At least 16 points.
> - **Equipment:** Only blank paper and pen – no calculators.

## Instructions

- Each task must be solved on paper using pen.  
- Write your name, Neptun code, and teacher’s name on the top of this test paper; also write your name and Neptun code on every page you use.  
- After finishing, place all solution papers behind this test paper and fold them in half (parallel to the longer side) so they stay together.  
- This test paper will be shared on Canvas.  

> [!warning] Justification required
> 
> Justify your answers to **every question except Question 1** — brief justifications suffice. A *yes/no* answer without justification scores **0 marks** (except Q1).

> [!tip] Leave it unevaluated
> 
> You may leave binomial coefficients or factorials as-is — no calculator is needed.

> [!info]  
> 
> In **Question 6** you have a choice between two versions. Solve only one of them.

---

## Questions

### 1. (7 marks)

*Proofs are not required – only the answers.*

1. In a running race there are $24$ participants. How many different outcomes are possible for the first $5$ places?  
2. In how many different ways can $5$ identical gifts be distributed among $12$ people, if each person can get at most one gift?  
3. How many different strings can be formed using exactly the following letters:  
   $A, A, A, B, C, C, D, D, D$?  
4. In how many different ways can $12$ different gifts be distributed among $7$ people, if anyone can get any number of gifts?  
5. In how many different ways can $20$ people sit down around a round table? (Rotations are considered identical.)  
6. In how many different ways can $8$ identical apples be distributed among $11$ children?  
7. At least how many people do we need to have in a group to be **certain** that there are $3$ among them who were born in the same month of the year?  

---

### 2. (3 marks)

The administration office at the Faculty of Informatics is planning to purchase **100 mouse pads** in total. There are four different types.

1. In how many different ways can they choose the $100$ mouse pads? (Types are distinguished, pads of the same type are identical.)  
2. In how many ways if they order **at least $5$ mouse pads of each type**?  

---

### 3. (7 marks)

Boris has $10$ books and puts them all on one shelf in a row. Three of them are:  
*War and Peace* (W), *Pride and Prejudice* (P), *Crime and Punishment* (C).

1. In how many ways can he arrange the $10$ books so that **W and P are not next to each other**?  
2. In how many ways so that **W, P, and C are next to each other** (in some order)?  
3. In how many ways so that **W is the first book** and **C is not the last book**?  

---

### 4. (9 marks)

Consider 7-digit numbers formed exactly from the digits:  
$0, 1, 2, 3, 4, 4, 5$.

1. How many such 7-digit numbers exist?  
2. How many of them are divisible by $5$?  
3. How many have the **last digit not equal to $4$**?  

---

### 5. (7 marks)

1. In the expansion of  
   $$
   \left(4x^{5} - \frac{2}{x^{3}}\right)^{10}
   $$  
   find the coefficients of the terms $x^{50}$ and $x^{27}$.

2. A school has $150$ students. Activities: chess (C), volleyball (V), badminton (B).  
   Given:  
   $|C| = 45,\ |V| = 54,\ |B| = 72,$  
   $|C \cap V| = 17,\ |C \cap B| = 26,\ |V \cap B| = 21,$  
   $37$ students do none of these.  

   How many take part in **all three** activities?  

---

## Question 6 (choose only one version)

### Version 1 (7 marks)

After the exam, $20$ DM1 students attend a dance.

1. In how many ways can they form **one circle** so that Anna and Bella stand next to each other, but Anna does **not** stand next to Celia?  
2. Exactly $10$ men and $10$ women. In how many ways can they form **one circle** so that men and women alternate (no two men or two women are adjacent)?  

---

### Version 2 (7 marks)

For each sequence below, decide if there exists  
(1) a **graph**  
(2) a **simple graph**  
with the given degree sequence. If yes, draw an example; if not, prove impossibility.

1. $5,4,3,3,3,2,2$  
2. $6,4,4,4,4,2,0$  
3. $6,5,4,4,3,2,1$  

---