Let’s refresh the definitions. “A permutation of a set of distinct objects is an ordered arrangement of these objects. We also are interested in ordered arrangements of some of the elements of a set. An ordered arrangement of r elements of a set is called an r-permutation. ” “An r-combination of elements of a set […]
First, let’s examine the definitions of permutations and combinations. “A permutation of a set of distinct objects is an ordered arrangement of these objects. We also are interested in ordered arrangements of some of the elements of a set. An ordered arrangement of r elements of a set is called an r-permutation. ” “An r-combination
In how many different orders can five runners finish a race if no ties are allowed? Read More »
This is a simple exercise. To solve it, we just need to apply the formula to calculate the number of r-combinations. “The number of r-combinations of a set with n elements, where n is a nonnegative integer and r is an integer with 0 ≤ r ≤ n, equals C(n,r) = n! / r!(n−r)!” Discrete
This is another simple exercise. To solve it, we just need to apply the formula to calculate the number of 3-permutations. “If n and r are integers with 0 ≤ r ≤ n, then P(n,r) = n! / (n−r)! ”. Discrete mathematics and its applications by Rosen. Answer: We just need to calculate the number
Find the number of 5-permutations of a set with nine elements Read More »
This type of exercise is easy. You are only required to apply a formula. You just need to know the formula. “If n and r are integers with 0 ≤ r ≤ n, then P(n,r) = n! / (n−r)! ”. Discrete mathematics and its applications by Rosen. a) P(6,3) P(6,3)=6!/(6-3)! =6!/3! =6x5x4x3!/3! =120 b) P(6,5)
Find the value of each of these quantities Read More »
“A permutation of a set of distinct objects is an ordered arrangement of these objects.” We can follow a simple approach to solve this exercise. If we calculate how many permutations are there in the set {b, c, d, e, f, g} (the original set without ‘a’), we will know how many permutations of the
How many permutations of {a, b, c, d, e, f, g} end with a? Read More »
Relevant definitions: Definition: “A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a,b) ∈ R, for all a,b ∈ A. A relation R on a set A such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a =
Let’s refresh the definitions that are relevant to this exercise. Definition: “A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.” Definition: “A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a,b) ∈ R, for all a,b
Let’s start with relevant definitions. Definition: “A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.” Definition: “A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a,b) ∈ R, for all a,b ∈ A. A relation R
To list the ordered pairs, we must follow the condition stated in the relation. In this case, (a,b) ∊R if a divides b. In other words, when we divide b by a, we get as a result an integer value, and the remainder is 0. a) List all the ordered pairs in the relation R
