Endterm Sample Spring 2026

Discrete Mathematics - Endterm Sample - Spring 2026

NAME:
NEPTUN code:
Group number/Class Teacher:

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.

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).

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

---