Given N and K. The task is to find out how many different ways are there to represent N as the sum of K non-zero integers.
Input: N = 5, K = 3
The possible combinations of integers are:
( 1, 1, 3 )
( 1, 3, 1 )
( 3, 1, 1 )
( 1, 2, 2 )
( 2, 2, 1 )
( 2, 1, 2 )
Input: N = 10, K = 4
The approach to the problem is to observe a sequence and use combinations to solve the problem. To obtain a number N, N 1’s are required, summation of N 1’s will give N. The problem allows to use K integers only to make N.
Let's take N = 5 and K = 3, then all possible combinations of K numbers are: ( 1, 1, 3 ) ( 1, 3, 1 ) ( 3, 1, 1 ) ( 1, 2, 2 ) ( 2, 2, 1 ) ( 2, 1, 2 ) The above can be rewritten as: ( 1, 1, 1 + 1 + 1 ) ( 1, 1 + 1 + 1, 1 ) ( 1 + 1 + 1, 1, 1 ) ( 1, 1 + 1, 1 + 1 ) ( 1 + 1, 1 + 1, 1 ) ( 1 + 1, 1, 1 + 1 )
From above, a conclusion can be drawn that of N 1’s, k-1 commas have to be placed in between N 1’s and the remaining places are to be filled with ‘+’ signs. All combinations of placing k-1 commas and placing ‘+’ signs in the remaining places will be the answer. So, in general, for N there will be N-1 spaces between all 1, and out of those choose k-1 and place a comma in between those 1. In between the rest 1’s, place ‘+’ signs. So ways of choosing K-1 objects out of N-1 is . The dynamic programming approach is used to calculate .
Below is the implementation of the above approach:
Total number of different ways are 6
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
- Represent (2 / N) as the sum of three distinct positive integers of the form (1 / m)
- Ways to represent a number as a sum of 1's and 2's
- Number of ways to represent a number as sum of k fibonacci numbers
- Number of distinct ways to represent a number as sum of K unique primes
- Count of ways to represent N as sum of a prime number and twice of a square
- Count of integers up to N which represent a Binary number
- Count ways to express 'n' as sum of odd integers
- Count ways to express even number ‘n’ as sum of even integers
- Ways to form an array having integers in given range such that total sum is divisible by 2
- Number of ways to write N as a sum of K non-negative integers
- Different ways to sum n using numbers greater than or equal to m
- Count of different ways to express N as the sum of 1, 3 and 4
- Represent a number as a sum of maximum possible number of Prime Numbers
- Represent N as sum of K even numbers
- Number of ways to form a heap with n distinct integers
- Ways to select one or more pairs from two different sets
- Ways of dividing a group into two halves such that two elements are in different groups
- Calculate Stirling numbers which represents the number of ways to arrange r objects around n different circles
- Number of ways to arrange K different objects taking N objects at a time
- Number of arrays of size N whose elements are positive integers and sum is K
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.