Let’s start by reviewing the relevant definitions needed to solve this type of exercise. Definition: “Let A and B be sets. A binary relation from A to B is a subset of A × B.” Definition: “A relation R on a set A is called reflexive if (a,a) ∈ R for every element a ∈ […]
Definition: “Let A and B be sets. A binary relation from A to B is a subset of A × B.” Discrete Mathematics and its Applications by Rosen. From the definition above, we can easily see that in a binary relation from A to B, the first element will belong to A and the second
A powerful tool for solving problems is to transform a given problem into another one that is easier to solve, or that we already know how to solve. Then, from the solution to that new problem, we can give the solution to the former one. This is such a case. We know that we have
“A permutation of a set of distinct objects is an ordered arrangement of these objects.” Theorem: “If n is a positive integer and r is an integer with 1 ≤ r ≤ n, then there are P (n, r ) = n(n − 1)(n − 2) · · · (n − r + 1) r-permutations
Let S = {1,2,3,4,5}. List all the 3-permutations of S. List all the 3-combinations of S. Read More »
This type of exercise is quite easy, and probably you won’t find it commonly on a test, exam, or in real life. However, there are some places you might find it. So, it is worth looking into it. As usual, let’s start with relevant definitions and theorems. “A permutation of a set of distinct objects
How many different permutations are there of the set {a,b,c,d,e,f,g}? Read More »
A gentle reminder. “A permutation of a set of distinct objects is an ordered arrangement of these objects.” Theorem: “If n is a positive integer and r is an integer with 1 ≤ r ≤ n, then there are P (n, r ) = n(n − 1)(n − 2) · · · (n − r
How many permutations of the letters ABCDEFG contain Read More »
How to address this question? First, we need to see that we are asked: “in how many ways a set can be selected”. As per the definition of a set, we should know that the elements in a set don’t have an order. In other words, a set is an unordered collection of elements. So,
In how many ways can a set of five letters be selected from the English alphabet? Read More »
Let’s examine the first question. How many bit strings of length 10 contain exactly four 1s? From the question, you can see that the order of the four 1s does not matter. We just need to find out how many bit strings of length 10 contain exactly four 1s. This reminds me of a definition.
How many bit strings of length 10 contain: Read More »
Before starting to solve the exercise, let’s start with the definitions. “A permutation of a set of distinct objects is an ordered arrangement of these objects.” It is also important to know beforehand how many of these ordered arrangements are there. So, we don’t forget anyone when asked this type of question. Theorem: “If n
Like in other exercises, we should start with the definitions. “A permutation of a set of distinct objects is an ordered arrangement of these objects.” It is also important to know beforehand how many of these ordered arrangements are there, so we don’t forget anyone when asked this type of question. Theorem: “If n is
List all the permutations of {a, b, c} Read More »
