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# Category Archives: Discrete Mathematics and Combinatorics

## Generalized Permutations & Combinations problems – Discrete Math & Combinatorics

**Permutations with Repetition example question**

How many ways are there to assign three jobs to five employees if each employee can be given more than one job?

Solution to this **Discrete Math** practice problem is given in the video below!

**Combinations with Repetition example problem**

How many ways are there to select three unordered elements from a set with five elements when repetition is allowed?

Solution to this **Discrete Math** practice problem is given in the video below!

**Combinations with Repetition example**

A book publisher has 3,000 copies of a discrete mathematics book. How many ways are there to store these books in their three warehouses if the copies of the book are indistinguishable?

Solution to this **Discrete Math** practice problem is given in the video below!

**Combinations with Repetition HARD example question**

How many positive integers less than 1,000,000 have the sum of their digits equal to 19?

Solution to this **Discrete Math** practice problem is given in the video below!

**Combinations with Repetition HARD example problem**

How many solutions are there to the equation *x*_{1} + *x*_{2} + *x*_{3} + *x*_{4} + *x*_{5} = 21, where *x _{i }*,

*i*= 1, 2, 3, 4, 5, is a nonnegative integer such that

a)

*x*

_{1}≥ 1 ?

b)

*x*≥ 2 for

_{i}*i*= 1, 2, 3, 4, 5 ?

c) 0 ≤

*x*

_{1}≤ 10 ?

d) 0 ≤

*x*

_{1}≤ 3, 1 ≤

*x*

_{2}< 4, and

*x*

_{3}≥ 15 ?

Solution to this **Discrete Math** practice problem is given in the video below!

**Combinations with Repetition HARD example**

There are 10 questions on a discrete mathematics final exam. How many ways are there to assign scores to the problems if the sum of the scores is 100 and each questions is worth at least 5 points?

Solution to this **Discrete Math** practice problem is given in the video below!

**Distinguishable Objects & Distinguishable Boxes example question**

How many ways are there to distribute 12 distinguishable objects into six distinguishable boxes so that two objects are placed in each box?

Solution to this **Discrete Math** practice problem is given in the video below!

## Pigeonhole Principle problems – Discrete Math

**Pigeonhole Principle example question**

a) Show that if five integers are selected from the first eight positive integers, there must be a pair of these integers with a sum equal to 9.

b) Is the conclusion in part (a) true if four integers are selected rather than five?

Solution to this **Discrete Math** practice problem is given in the video below!

**Pigeonhole Principle example problem**

How many numbers must be selected from the set {1,2,3,4,5,6} to guarantee that at least one pair of these numbers add up to 7?

Solution to this **Discrete Math** practice problem is given in the video below!

**Generalized Pigeonhole Principle example**

A drawer contains a dozen brown socks and a dozen black socks, all unmatched. A man takes socks out at random in the dark.

a) How many socks must be he take out to be sure that he has at least two socks of the same color?

b) How many socks must he take out to be sure that he has at least two black socks?

Solution to this **Discrete Math** practice problem is given in the video below!

**Generalized Pigeonhole Principle example question**

What is the minimum number of students, each of whom comes from one of the 50 states, who must be enrolled in a university to guarantee that there are at least 100 who come from the same state?

Solution to this **Discrete Math** practice problem is given in the video below!

**Generalized Pigeonhole Principle example problem**

A company stores public products in a warehouse. Storage bins in this warehouse are specified by their aisle, location in the aisle, and shelf. There are 50 aisles, 85 horizontal locations in each aisle, and 5 shelves throughout the warehouse. What is the least number of products the company can have so that at least two products must be stored in the same bin?

Solution to this **Discrete Math** practice problem is given in the video below!

**Generalized Pigeonhole Principle example #4**

There are 38 different time periods during which classes at a university can be scheduled. If there are 677 different classes, how many different rooms will be needed?

Solution to this **Discrete Math** practice problem is given in the video below!

**RAMSEY THEORY Generalized Pigeonhole Principle example question**

Show that in a group of 10 people (where any two people are either friends or enemies), there are either three mutual friends or four mutual enemies, and there are either three mutual enemies or four mutual friends.

Solution to this **Discrete Math** practice problem is given in the video below!

## Basics of Counting problems in Discrete Math

**Basics of Counting example question**

How many positive integers between 50 and 100

a) Are divisible by 7? What are these integers?

b) Are divisible by 11? What are these integers?

c) Are divisible by both 7 and 11? What are these integers?

Solution to this **Discrete Math** practice problem is given in the video below!

**Basics of Counting ****example problem #2**

How many positive integers between 100 and 999 inclusive

a) Are divisible by 7?

b) Are odd?

c) Have the same three decimal digits?

d) Are NOT divisible by 4?

e) Are divisible by 3 or 4?

f) Are NOT divisible by either 3 or 4?

g) Are divisible by 3 but NOT 4?

h) Are divisible by 3 and 4?

Solution to this **Discrete Math** practice problem is given in the video below!

**Basics of Counting ****example #3**

How many strings of three decimal digits

a) Do NOT contain the same digit three times?

b) Begin with an odd digit?

c) Have exactly two digits that are 4s?

Solution to this **Discrete Math** practice problem is given in the video below!

**Basics of Counting ****example question #4**

How many functions are there from the set {1, 2, …, *n*}, where *n* is a positive integer, to the set {0, 1}

a) That are one-to-one?

b) That assign 0 to both 1 and *n *?

c) That assign 1 to exactly one of the positive integers less than *n *?

Solution to this **Discrete Math** practice problem is given in the video below!

**Basics of Counting ****example problem #5**

How many ways are there to seat six people around a circular table where two seatings are considered the same when everyone has the same two neighbors without regard to whether they are right or left neighbors?

Solution to this **Discrete Math** practice problem is given in the video below!

## Discrete Mathematics & Combinatorics problems (complete Playlist)

Get help with many different examples and practice problems in **Discrete Mathematics** that are applicable to Probability, Electrical Engineering, Computer Science, and other courses.

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