Given two integer N and R, the task is to calculate the number of ways to distribute N identical objects into R distinct groups such that no groups are left empty.
Input: N = 4, R = 2
No of objects in 1st group = 1, in second group = 3
No of objects in 1st group = 2, in second group = 2
No of objects in 1st group = 3, in second group = 1
Input: N = 5, R = 3
Approach: Idea is to use Multinomial theorem. Let us suppose that x1 objects are placed in the first group, x2 objects are placed in second group and xR objects are placed in the Rth group. It is given that,
x1 + x2 + x3 +…+ xR = N for all xi ≥ 1 for 1 ≤ i ≤ R
Now replace every xi with yi + 1 for all 1 ≤ i ≤ R. Now all the y variaables are greater than or equal to zero.
The equation becomes,
y1 + y2 + y3 + … + yR + R = N for all yi ≥ 0 for 1 ≤ i ≤ R
y1 + y2 + y3 + … + yR = N – R
It now reduces to that standard multinomial equation whose solution is (N – R) + R – 1CR – 1.
The solution of this equation is given by N – 1CR – 1.
Below is the implementation of the above approach:
Time Complexity: O(R)
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.
- Number of ways of distributing N identical objects in R distinct groups
- Number of ways to arrange K different objects taking N objects at a time
- Distributing M items in a circle of size N starting from K-th position
- Probability of distributing M items among X bags such that first bag contains N items
- Calculate Stirling numbers which represents the number of ways to arrange r objects around n different circles
- Divide Matrix into K groups of adjacent cells having minimum difference between maximum and minimum sized groups
- Count the number of ways to divide N in k groups incrementally
- Number of distinct ways to represent a number as sum of K unique primes
- Ways of dividing a group into two halves such that two elements are in different groups
- Check if given two straight lines are identical or not
- Minimum area of square holding two identical rectangles
- Find side of Square which makes minimal area to fit two identical rectangles inside it
- Count the number of ways to fill K boxes with N distinct items
- Count ways to split N! into two distinct co-prime factors
- Maximum number of objects that can be created as per given conditions
- Number of Groups of Sizes Two Or Three Divisible By 3
- Maximize number of groups formed with size not smaller than its largest element
- Maximize count of empty water bottles from N filled bottles
- Sum of values of all possible non-empty subsets of the given array
- Product of values of all possible non-empty subsets of given Array
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. 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.