Given two natural number n and m. The task is to find the number of ways in which the numbers that are greater than or equal to m can be added to get the sum n.
Input : n = 3, m = 1 Output : 3 Following are three different ways to get sum n such that each term is greater than or equal to m 1 + 1 + 1, 1 + 2, 3 Input : n = 2, m = 1 Output : 2 Two ways are 1 + 1 and 2
The idea is to use Dynamic Programming by define 2D matrix, say dp. dp[i][j] define the number of ways to get sum i using the numbers greater than or equal to j. So dp[i][j] can be defined as:
If i < j, dp[i][j] = 0, because we cannot achieve smaller sum of i using numbers greater than or equal to j.
If i = j, dp[i][j] = 1, because there is only one way to show sum i using number i which is equal to j.
Else dp[i][j] = dp[i][j+1] + dp[i-j][j], because obtaining a sum i using numbers greater than or equal to j is equal to the sum of obtaining a sum of i using numbers greater than or equal to j+1 and obtaining the sum of i-j using numbers greater than or equal to j.
Below is the implementation of this approach:
- Number of ways to divide an array into K equal sum sub-arrays
- Count number of binary strings such that there is no substring of length greater than or equal to 3 with all 1's
- Number of ways of choosing K equal substrings of any length for every query
- Ways of filling matrix such that product of all rows and all columns are equal to unity
- Number of ways to get even sum by choosing three numbers from 1 to N
- Count number of ways to arrange first N numbers
- Bell Numbers (Number of ways to Partition a Set)
- Calculate Stirling numbers which represents the number of ways to arrange r objects around n different circles
- Count numbers whose difference with N is equal to XOR with N
- Count numbers (smaller than or equal to N) with given digit sum
- Minimum number greater than the maximum of array which cannot be formed using the numbers in the array
- Count numbers in given range such that sum of even digits is greater than sum of odd digits
- Number of ways to represent a number as sum of k fibonacci numbers
- Count of Numbers in Range where first digit is equal to last digit of the number
- Count of subarrays whose maximum element is greater than k
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