Determine the number of different ways to partition a set of n elements into k clusters
Determine the number of different ways to partition a set of n elements into k clusters
Equivalent Relations.. a-) Determine the number of different equivalence relations on a set with 4 elements. b-) Generalize your answer to part (a) for a set with n elements.
Hierarchical clustering is sometimes used to generate K clusters, K > 1 by taking the clusters at the Kth level of the dendrogram. (Root is at level 1.) By looking at the clusters produced in this way, we can evaluate the behavior of hierarchical clustering on different types of data and clusters, and also compare hierarchical approaches to K-means. The following is a set of one-dimensional points: {6, 12, 18, 24, 30, 42, 48}. (a) For each of the following...
Find out the number of ways of dividing n different chocolates to 3 children such that the each of them gets a, b and c chocolates respectively by coming up with a k-to-1 function between the set of all permutations of the n chocolates and the set of all distinct ways of dividing the chocolates among the 3 children.
Prove that the number of unordered sequences of length k with elements from a set X of size n is n+k−1 k . Hint: For illustration, first consider the example n = 4, k = 6. Let the 4 elements of the set X be denoted a, b, c, d. Argue that any unordered sequence of size 6 consisting of elements a, b, c, d can be represented uniquely by a symbol similar to “··|·|··|·”, corresponding to the sequence aabccd....
Please write full justification for (a) and (b). Will uprate/vote! 4. K-means The goal of K-means clustering is to divide a set of n points into k< n subgroups of points that are "close" to each other. Each subgroup (or cluster) is identified by the center of the cluster, the centroid (μι, μ2' ··· ,14k) In class, we have seen a brute force approach to solve this problem exactly. Each of the k clusters is represented by a color, e.g.,...
Let A be a set with m elements and B be a set with n elements in it. -When is it possible to have a k-to-1 function f such that f : A → B? -Count the number of k-to-1 functions f such that f : A → B
Find a rec. relation for an,k, the number of ways to order n doughnuts from k different types of doughnuts if two or four or six doughnuts must be chosen of each type. Answer is an,k = an-2,k-1+an-4,k-1+ an-6,k-1., please explain how to geit it, thanks.
7. Let A, , An be non-empty subsets of a finite set Ω. If 1 k n and Ek is the set of elements in Ω which belong to at least k of the Ai's show that Pal i-1 7. Let A, , An be non-empty subsets of a finite set Ω. If 1 k n and Ek is the set of elements in Ω which belong to at least k of the Ai's show that Pal i-1
Find the generating function to determine the number of ways to pick k objects from n objects when the ith object can appear times for and any integer . i+jn u>ι>Ι We were unable to transcribe this image
Let A be a set with m elements and B a set of n elements, where m, n are positive integers. Find the number of one-to-one functions from A to B.