Suppose we have a set of n identical balls in a bag. How many ways can m players choose balls from the bag such that the first player gets at least k balls. Assume k< n.
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- Consider 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.A drawer contains 7 pair of socks where 8 socks are white, and 6 are red. In how many ways we can take 14 random socks out of the drawer so each sock is next to its matching pair. a. 441 ways b. 42 ways c. 1225 ways d. 70 waysLet n be a natural number. Let's take the following 2 player game. We have .n matches. Player 1 takes one or two matches from the pile, then as long as there are still matches, player 2 takes one or two matches, and so on, alternating until there are no more matches left. The player who takes the last match loses. The losing player gets a benefit of -1 and the other player gets a benefit of 1. Represent this game for n=2 ,n=3 and n=4 and say how many strategies each player has.
- Consider a tournament between N teams, each team playing each of the other teams. Show (by example) there is a tournament that might occur, where every team is beaten by some team.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:Imagine there are N teams competing in a tournament, and that each team plays each of the other teams once. If a tournament were to take place, it should be demonstrated (using an example) that every team would lose to at least one other team in the tournament.
- A saleswoman wants to visit eastern cities New York City, Boston, Cleveland, and Washington, D.C. Driving distances are as shown in the figure below. what is the shortest trip starting in New York that visits each of these cities, if r1=493, r 2 = 560 ? Find the solution using the sorted-edge method.Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you ll in the values is the correct one.
- Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Prove that the coin changing problem exhibits optimal substructure. Design a recursive backtracking (brute-force) algorithm that returns the minimum number of coins needed to make change for n cents for any set of k different coin denominations. Write down the pseudocode and prove that your algorithm is correct.There are n people who want to carpool during m days. On day i, some subset si ofpeople want to carpool, and the driver di must be selected from si . Each person j hasa limited number of days fj they are willing to drive. Give an algorithm to find a driverassignment di ∈ si each day i such that no person j has to drive more than their limit fj. (The algorithm should output “no” if there is no such assignment.) Hint: Use networkflow.For example, for the following input with n = 3 and m = 3, the algorithm could assignTom to Day 1 and Day 2, and Mark to Day 3. Person Day 1 Day 2 Day 3 Limit 1 (Tom) x x x 2 2 (Mark) x x 1 3 (Fred) x x 0Given is a list of K distinct coin denominations (V1,...,Vk) and the total sum S>0. Find the minimum number of coins whose sum is equal to S (we can use as many coins of one type as we want), or report that it’s not possible to select coins in such a way that they sum up to S. Justify your explanation!