(a)
To argue that numbers of ways of placing the balls in bins is
(a)
Explanation of Solution
Given information:
The n balls are distinct and their order within bin doesn’t matter.
Explanation:
There can be b different decisions made for n balls about their placement. The total number of possibilities is just
(b)
To prove that there are exactly
(b)
Explanation of Solution
Given information:
It is assumed that balls are distinct and that balls in each bin are ordered.
Explanation:
First assume that sticks can be distinguished. This implies that there are total of n balls and
This arrangement can be related with the original statement, where sticks can be imagined as dividing lines between bins and ordered balls between them can be imagined as ordered balls in each bin.
(c)
To show that
(c)
Explanation of Solution
Given information:
The balls are identical and their order within a bin does not matter.
Explanation:
Using results from above two parts, it can be noticed that any of the n permutation of balls will result in the similar configuration. Thus, count from the previous parts must be divided by
(d)
To show that number of ways of placing the balls is
(d)
Explanation of Solution
Given information:
The balls are identical and no bin may contain more than one ball.
Explanation:
Here, a set of bins to contain balls is selected,as each bin can have a ball or not. The numbers of bins selected is n since number of non-empty bins and the numbers of balls must be equal. In other words, a subset of size n of the bins is being selected from the whole set of bins. This becomes the combinatorial definition of
(e)
To show that number of ways of placing the balls is
(e)
Explanation of Solution
Given information:
The balls are identical and no bin may be left empty.
Explanation:
The condition is to put one ball in each bin as no bin can be left empty. Thus there are
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Chapter C Solutions
Introduction to Algorithms
- Suppose a set of n objects is given, such that the size of the ith object satisfies that 0 < < 1. It is required to pack all the objects in a set of boxes of unit size. Each box can contain any subset of objects whose total size does not exceed 1. The heuristic first-fit takes each object and fits it into the first box that can hold it. Finally, let S be sum of the sizes of all objects: a. Show that the total number of boxes used by the first-fit heuristic is never greater than ⌈2S⌉. b. Show that the approximation factor of the first-fit heuristic is 2.arrow_forwardConsider the challenge of determining whether a witness questioned by a law enforcement agency is telling the truth. An innovative questioning system pegs two individuals against each other. A reliable witness can determine whether the other individual is telling the truth. However, an unreliable witness's testimony is questionable. Given all the possible outcomes from the given scenarios, we obtain the table below. This pairwise approach could then be applied to a larger pool of witnesses. Answer the following: 1) If at least half of the K witnesses are reliable, the number of pairwise tests needed is Θ(n). Show the recurrence relation that models the problem. Provide a solution using your favorite programming language, that solves the recurrence, using initial values entered by the user.arrow_forwardA triomino is an L-shaped tile formed by 1-by-1 adjacent squares. Given an 2^n × 2^n board (n ≥ 1) with one missing square, it is always possible to cover the board by triominos without overlapping tiles (you may assume this fact, or consider your solution to this question as a constructive proof) Given an 2^n×2^n board with one missing square, design a divide-and-conquer algorithm to determine how to cover the board with triominos.arrow_forward
- Assume we have two groups A and B of n cups each, where group A has n black cups while group B has n white cups. The cups in both groups have different shapes and hence a different amount of coffee per each cup. Given the following two facts: 1) All black cups hold different amounts of coffee, 2) Each black cup has a corresponding white cup that holds exactly the same amount of coffee, your task is to find a way to group the cups into pairs of black and white cups that hold the same amount of coffee. For example:arrow_forwardCorrect answer will be upvoted else downvoted. Computer science. way from block u to obstruct v is a grouping u=x0→x1→x2→⋯→xk=v, where there is a street from block xi−1 to hinder xi for each 1≤i≤k. The length of a way is the amount of lengths over all streets in the way. Two ways x0→x1→⋯→xk and y0→y1→⋯→yl are unique, if k≠l or xi≠yi for some 0≤i≤min{k,l}. Subsequent to moving to another city, Homer just recalls the two exceptional numbers L and R yet fails to remember the numbers n and m of squares and streets, separately, and how squares are associated by streets. Be that as it may, he accepts the number of squares ought to be no bigger than 32 (in light of the fact that the city was little). As the dearest companion of Homer, if it's not too much trouble, let him know whether it is feasible to see as a (L,R)- constant city or not. Input The single line contains two integers L and R (1≤L≤R≤106). Output In case it is difficult to track down a (L,R)- consistent city…arrow_forwardSolve the problems below using the pigeonhole principle: A) How many cards must be drawn from a standard 52-card deck to guarantee 2 cardsof the same suit? Note that there are 4 suits. B) Prove that if four numbers are chosen from the set {1, 2, 3, 4, 5, 6}, at least onepair must add up to 7.Hint: Find all pairs of numbers from the set that add to 7.C) Prove that for any 10 given distinct positive integers that are less than 100, thereexist two different non-empty subsets of these 10 numbers, whose members have the samesum.An example of the 10 given numbers could be 23, 26, 47, 56, 14, 99, 94, 78, 83, 69. Onesubset of the 10 numbers could be {23, 26, 47, 56}, and another subset could be {83, 69}.The sum of the elements in the first set is 152, and it is equal to the sum of the elements inthe second subset.Hint: identify how many pigeons and how many holes you have before using the pigeonholeprinciple.arrow_forward
- The rook is a chess piece that may move any number of spaces either horizontally or vertically. Consider the “rooks problem” where we try to place 8 rooks on an 8x8 chess board in such a way that no pair attacks each other. a. How many different solutions are there to this?b. Suppose we place the rooks on the board one by one, and we care about the order in which we put them on the board. We still cannot place them in ways that attack each other. How many different full sequences of placing the rooks (ending in one of the solutions from a) are there?arrow_forward6. Consider a binary classification problem using 1-nearest neighbors with the Euclidean distance metric. We have N 1-dimensional training points x(1), x(2), . . . x(N ) and corresponding labelsy(1), y(2), . . . y(N ) with x(i ) ∈ R and y(i ) ∈ {0, 1}. Assume the points x(1), x(2), . . . x(N ) are in ascending order by value. If there are ties during the 1-NN algorithm, we break ties by choosing the labelcorresponding to the x(i ) with lower value.arrow_forwardConsider an N-team tournament in which each team plays every other team once. If a tournament were to materialise, show (by example) that every team would be defeated by at least one other team.arrow_forward
- 1. Let T (n) be the number of moves in our solution to the n-disc Towers ofHanoi puzzle. Recall that to solve this puzzle,•move the top n −1 discs from the source to the scratch peg,•move disc n from the source to the destination peg, and•move the top n −1 discs from the scratch to the destination peg.The first step takes T (n −1) moves, the second step takes one move, andthe third step takes T (n −1) moves again. In other words,T (n) = 2T (n −1) + 1.This applies when n > 0. At zero, we have T (0) = 0, because with zerodiscs the start and final states are identical; there are no moves to make.Your task is to find a closed-form expression for T (n) (i.e., one that doesnot use recursion), and prove that it’s correct using induction.arrow_forwardRequire the solution asap only do if you know the answers any error will get you downvoted for sure.arrow_forwardCorrect answer will be upvoted else Multiple Downvoted. Don't submit random answer. Computer science. You are given an integer k and n particular focuses with integer facilitates on the Euclidean plane, the I-th point has arranges (xi,yi). Consider a rundown of all the n(n−1)2 sets of focuses ((xi,yi),(xj,yj)) (1≤i<j≤n). For each such pair, work out the separation from the line through these two focuses to the beginning (0,0). You will likely work out the k-th most modest number among these distances. Input The principal line contains two integers n, k (2≤n≤105, 1≤k≤n(n−1)2). The I-th of the following n lines contains two integers xi and yi (−104≤xi,yi≤104) — the directions of the I-th point. It is ensured that all given focuses are pairwise particular. Output You should output one number — the k-th littlest separation from the beginning. Your answer is considered right if its outright or relative blunder doesn't surpass 10−6. Officially, let your answer be…arrow_forward
Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole