Given two positive integers N and K. The task is to find the number of arrays of size N that can be formed such that elements of the array should be positive integers and the sum of elements is equal to K.
Input : N = 2, K = 3 Output : 2 Explanation: [1, 2] and [2, 1] are the only arrays of size 2 whose sum is 3. Input : n = 3, k = 7 Output : 15
Prerequisite: Stars and Bars
Suppose there are K identical objects which needs to be placed in N bins (N indices of the array) such that each bin have at least one object. Instead of starting to place objects into bins, we start placing the objects on a line, where the object for the first bin will be taken from the left, followed by the objects for the second bin, and so forth. Thus, the configuration will be determined once one knows what is the first object going to the second bin, and the first object going to the third bin, and so on. We can indicate this by placing N X 1 separating bars at some places between two objects; since no bin is allowed to be empty, there can be at most one bar between a given pair of objects. So, we have K objects in a line with K – 1 gaps. Now we have to choose N – 1 gaps to place bars from K – 1 gaps. This can be chosen by K – 1CN – 1.
Below is implementation of this approach:
- Find the number of positive integers less than or equal to N that have an odd number of digits
- Number of ways in which N can be represented as the sum of two positive integers
- Check whether a number can be represented as sum of K distinct positive integers
- Maximum number of distinct positive integers that can be used to represent N
- Minimum product of k integers in an array of positive Integers
- Sum of product of all elements of sub-arrays of size k
- Minimum number of changes such that elements are first Negative and then Positive
- Ways to write n as sum of two or more positive integers
- Find K distinct positive odd integers with sum N
- Ways to write N as sum of two or more positive integers | Set-2
- Represent (2 / N) as the sum of three distinct positive integers of the form (1 / m)
- Longest sequence of positive integers in an array
- Check whether product of integers from a to b is positive , negative or zero
- Find n positive integers that satisfy the given equations
- Find all the possible remainders when N is divided by all positive integers from 1 to N+1
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