So, repeat the process above by multiplying 25 by 3, try to obtain combinations for sum 25, and you are done with this one. When I talk to students of mine over at Byte by Byte, nothing quite strikes fear into their hearts like dynamic programming. 2. Dynamic programming (DP) is as hard as it is counterintuitive. My version of this problem is a little different. Bitmasking and Dynamic Programming | Set 1, Bitmasking and Dynamic Programming | Set-2 (TSP), Bell Numbers (Number of ways to Partition a Set), Perfect Sum Problem (Print all subsets with given sum), Print Fibonacci sequence using 2 variables, Count even length binary sequences with same sum of first and second half bits, Sequences of given length where every element is more than or equal to twice of previous, LCS (Longest Common Subsequence) of three strings, Maximum product of an increasing subsequence, Count all subsequences having product less than K, Maximum subsequence sum such that no three are consecutive, Longest subsequence such that difference between adjacents is one, Maximum length subsequence with difference between adjacent elements as either 0 or 1, Maximum sum increasing subsequence from a prefix and a given element after prefix is must, Maximum sum of a path in a Right Number Triangle, Maximum sum of pairs with specific difference, Maximum size square sub-matrix with all 1s, Maximum number of segments of lengths a, b and c, Recursively break a number in 3 parts to get maximum sum, Maximum value with the choice of either dividing or considering as it is, Maximum weight path ending at any element of last row in a matrix, Maximum sum in a 2 x n grid such that no two elements are adjacent, Maximum difference of zeros and ones in binary string | Set 2 (O(n) time), Maximum path sum for each position with jumps under divisibility condition, Maximize the sum of selected numbers from an array to make it empty, Maximum subarray sum in an array created after repeated concatenation, Maximum path sum that starting with any cell of 0-th row and ending with any cell of (N-1)-th row, Minimum cost to fill given weight in a bag, Minimum sum of multiplications of n numbers, Minimum removals from array to make max – min <= K, Minimum steps to minimize n as per given condition, Minimum number of edits ( operations ) require to convert string 1 to string 2, Minimum time to write characters using insert, delete and copy operation, Longest Common Substring (Space optimized DP solution), Sum of all substrings of a string representing a number | Set 1, Find n-th element from Stern’s Diatomic Series, Find maximum possible stolen value from houses, Find number of solutions of a linear equation of n variables, Count number of ways to reach a given score in a game, Count ways to reach the nth stair using step 1, 2 or 3, Count of different ways to express N as the sum of 1, 3 and 4, Count ways to build street under given constraints, Counting pairs when a person can form pair with at most one, Counts paths from a point to reach Origin, Count of arrays having consecutive element with different values, Count ways to divide circle using N non-intersecting chords, Count the number of ways to tile the floor of size n x m using 1 x m size tiles, Count all possible paths from top left to bottom right of a mXn matrix, Count number of ways to fill a “n x 4” grid using “1 x 4” tiles, Size of array after repeated deletion of LIS, Remove array end element to maximize the sum of product, Convert to Strictly increasing array with minimum changes, Longest alternating (positive and negative) subarray starting at every index, Ways to sum to N using array elements with repetition allowed, Number of n-digits non-decreasing integers, Number of ways to arrange N items under given constraints, Probability of reaching a point with 2 or 3 steps at a time, Value of continuous floor function : F(x) = F(floor(x/2)) + x, Number of decimal numbers of length k, that are strict monotone, Different ways to sum n using numbers greater than or equal to m, Super Ugly Number (Number whose prime factors are in given set), Unbounded Knapsack (Repetition of items allowed), Print equal sum sets of array (Partition problem) | Set 1, Print equal sum sets of array (Partition Problem) | Set 2, Travelling Salesman Problem | Set 1 (Naive and Dynamic Programming), Longest palindrome subsequence with O(n) space, Count All Palindromic Subsequence in a given String, Count All Palindrome Sub-Strings in a String | Set 1, Number of palindromic subsequences of length k, Count of Palindromic substrings in an Index range, Count distinct occurrences as a subsequence, Longest Common Increasing Subsequence (LCS + LIS), LCS formed by consecutive segments of at least length K, Printing Maximum Sum Increasing Subsequence, Count number of increasing subsequences of size k, Printing longest Increasing consecutive subsequence, Construction of Longest Increasing Subsequence using Dynamic Programming, Find all distinct subset (or subsequence) sums of an array, Print all longest common sub-sequences in lexicographical order, Printing Longest Common Subsequence | Set 2 (Printing All), Non-decreasing subsequence of size k with minimum sum, Longest Common Subsequence with at most k changes allowed, Weighted Job Scheduling | Set 2 (Using LIS), Weighted Job Scheduling in O(n Log n) time, Minimum number of coins that make a given value, Collect maximum coins before hitting a dead end, Coin game winner where every player has three choices, Probability of getting at least K heads in N tosses of Coins, Count number of paths with at-most k turns, Count possible ways to construct buildings, Count number of ways to jump to reach end, Count number of ways to reach destination in a Maze, Count all triplets whose sum is equal to a perfect cube, Count number of binary strings without consecutive 1’s, Count number of subsets having a particular XOR value, Count Possible Decodings of a given Digit Sequence, Count number of ways to partition a set into k subsets, Count of n digit numbers whose sum of digits equals to given sum, Count ways to assign unique cap to every person, Count binary strings with k times appearing adjacent two set bits, Count of strings that can be formed using a, b and c under given constraints, Count digit groupings of a number with given constraints, Count all possible walks from a source to a destination with exactly k edges, Count Derangements (Permutation such that no element appears in its original position), Count total number of N digit numbers such that the difference between sum of even and odd digits is 1, Maximum difference of zeros and ones in binary string, Maximum and Minimum Values of an Algebraic Expression, Maximum average sum partition of an array, Maximize array elements upto given number, Maximum subarray sum in O(n) using prefix sum, Maximum sum subarray removing at most one element, K maximum sums of non-overlapping contiguous sub-arrays, Maximum Product Subarray | Added negative product case, Find maximum sum array of length less than or equal to m, Find Maximum dot product of two arrays with insertion of 0’s, Choose maximum weight with given weight and value ratio, Maximum sum subsequence with at-least k distant elements, Maximum profit by buying and selling a share at most twice, Maximum sum path in a matrix from top to bottom, Maximum decimal value path in a binary matrix, Finding the maximum square sub-matrix with all equal elements, Maximum points collected by two persons allowed to meet once, Maximum number of trailing zeros in the product of the subsets of size k, Minimum sum submatrix in a given 2D array, Minimum Initial Points to Reach Destination, Minimum Cost To Make Two Strings Identical, Paper Cut into Minimum Number of Squares | Set 2, Minimum and Maximum values of an expression with * and +, Minimum number of deletions to make a string palindrome, Minimum number of deletions to make a string palindrome | Set 2, Minimum jumps to reach last building in a matrix, Sub-tree with minimum color difference in a 2-coloured tree, Minimum number of deletions to make a sorted sequence, Minimum number of squares whose sum equals to given number n, Remove minimum elements from either side such that 2*min becomes more than max, Minimal moves to form a string by adding characters or appending string itself, Minimum steps to delete a string after repeated deletion of palindrome substrings, Clustering/Partitioning an array such that sum of square differences is minimum, Minimum sum subsequence such that at least one of every four consecutive elements is picked, Minimum cost to make Longest Common Subsequence of length k, Minimum cost to make two strings identical by deleting the digits, Minimum time to finish tasks without skipping two consecutive, Minimum cells required to reach destination with jumps equal to cell values, Minimum number of deletions and insertions to transform one string into another, Find if string is K-Palindrome or not | Set 1, Find if string is K-Palindrome or not | Set 2, Find Jobs involved in Weighted Job Scheduling, Find the Longest Increasing Subsequence in Circular manner, Find the longest path in a matrix with given constraints, Find the minimum cost to reach destination using a train, Find minimum sum such that one of every three consecutive elements is taken, Find number of times a string occurs as a subsequence in given string, Find length of the longest consecutive path from a given starting character, Find length of longest subsequence of one string which is substring of another string, Find longest bitonic sequence such that increasing and decreasing parts are from two different arrays, WildCard pattern matching having three symbols ( * , + , ? From Steps 2 and 3 above, you know all the double coin combinations that form a 100. The other coin remaining is 25. Given the weights and profits of ’N’ items, put these items in a knapsack which has a capacity ‘C’. Next, change your target sum to 100 - 2*25 = 50. (If it is not clear, try putting 5 and 25 for V1 and V2, for S = 100). This simple optimization reduces time complexities from exponential to polynomial. Although you are only demonstrating the approach with four coin values, you will see that attempting it with a considerably larger number of coins does not make it extremely complex.

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