Institute of Information Science - University of Miskolc

Combinatorics Exercises

An assembly plant wants to hire 5 male and 6 female workers. 6 men and 8 women apply for the job. How many ways can the desired workers be selected? (168)
A box contains 30 parts numbered from 1 to 30. How many ways can we select 6 parts such that among them there are 3 parts with specific numbers? (2,925)
A shipment contains 100 boxes: 75 are quality $A$ and 25 are quality $B$. How many ways can we select a 12-box sample in which the quality $B$ goods do not exceed 30%? (686,128,883,942,025)
12 students rent three boats: one with 3 seats, another with 4 seats, and the third with 5 seats. In how many ways can they be seated in the boats? (27,720)
How many five-digit numbers starting with 15 can be formed using the digits $1, 3, 5, 7, 9$, if each digit can be used only once? (6)
In how many ways can the letters of the word $MATHEMATICS$ be arranged (i.e., permutations of the word)? (4,989,600)
How many four-digit even numbers can be formed using the digits $0, 1, 2$ (with repetition allowed)? (36)
10 students qualified for the university sports day running competition. The first 3 will receive medals. How many different ways can the list of medalists be arranged? (720)
We roll a standard die four times in a row, and we write down the results of the rolls in order. How many different four-digit numbers can we obtain this way? (1,296)
How many three-digit numbers are there in which every digit is even? (100)
We will draw a total of 5 prizes among 15 students. A student can receive multiple prizes. How many different outcomes are possible for the drawing? (759,375)
How many subsets does a set with 6 elements have? (64)
7 people meet at a party, and everyone shakes hands with everyone else. How many handshakes are there in total? (21)
15 entries were submitted for a competition. 3 entries will be awarded, each with a 50 EURs. How many different ways can the prizes be awarded if they cannot be shared? (455)
How many subsets of the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ have at least 7 elements? (176)
We have fabric in $red$, $white$, $green$, and $blue$ colors, and we want to make flags. Each flag consists of horizontal stripes, and adjacent stripes cannot be the same color. How many different flags can we make if:
1. each flag must have two stripes? (12)
2. each flag must have three stripes? (36)
We placed a $red$, a $white$, and a $black$ dice in a box. We randomly draw two and roll them. How many possible outcomes are there for this experiment? (108)
After a graduation ceremony, each student received a photo from each of their classmates. How many students graduated if a total of 992 photos were exchanged? (32)