# Count of ways to distribute N items among 3 people with one person receiving maximum

Given an integer N, the task is to find the total number of ways to distribute N among 3 people such that:

• Exactly one person gets the maximum number of items among all the 3 people.
• Each person gets at least 1 item.

Examples:

Input: N = 5
Output: 3
Explanation:
3 way distribute the item among 3 people are {1, 1, 3}, {1, 3, 1} and {3, 1, 1}.
Distributions like {1, 2, 2} or {2, 1, 2} are not valid as two persons are getting the maximum.

Input: N = 10
Output: 33
Explanation:
For the Input N = 10 there are 33 ways of distribution.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:
To solve the problem mentioned above, we have to observe that if N < 4, then such a distribution is not possible.
For all values of N ≥ 4, follow the steps to solve the problem:

• Total no of ways to distribute N items among 3 people is given by (N – 1) * (N – 2) / 2.
• Initialize a variable s = 0 which stores the count of ways the distribution is not possible.
• Iterate two nested loops, where i ranges between [2, N – 3] and j ranging upto i:
• For each iteration, check if N = 2 * i + j, that is 2 persons can receive the maximum number of elements
• If so, then increment s by 1. If N is divisible by 3, update s by 3 * s + 1. Otherwise, update to 3 * s.
• Finally, return ans – s as the total number of ways to distribute N items among three people.

Below is the implementation of the above approach:

## C++

 `// C++ program to find the number ` `// of ways to distribute N item ` `// among three people such ` `// that one person always gets ` `// the maximum value ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Function to find the number ` `// of ways to distribute N ` `// items among 3 people ` `int` `countWays(``int` `N) ` `{ ` `    ``// No distribution ` `    ``// possible ` `    ``if` `(N < 4) ` `        ``return` `0; ` ` `  `    ``// Total number of ways to ` `    ``// distribute N items ` `    ``// among 3 people ` `    ``int` `ans = ((N - 1) * (N ` `                          ``- 2)) ` `              ``/ 2; ` ` `  `    ``// Store the number of ` `    ``// distributions which ` `    ``// are not possible ` `    ``int` `s = 0; ` ` `  `    ``for` `(``int` `i = 2; i <= N - 3; ` `         ``i++) { ` `        ``for` `(``int` `j = 1; j < i; ` `             ``j++) { ` ` `  `            ``// Count possiblities ` `            ``// of two persons ` `            ``// receiving the ` `            ``// maximum ` `            ``if` `(N == 2 * i + j) ` `                ``s++; ` `        ``} ` `    ``} ` ` `  `    ``// If N is divisible by 3 ` `    ``if` `(N % 3 == 0) ` `        ``s = 3 * s + 1; ` ` `  `    ``else` `        ``s = 3 * s; ` ` `  `    ``// Return the final ` `    ``// count of ways ` `    ``// to distribute ` `    ``return` `ans - s; ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``int` `N = 10; ` `    ``cout << countWays(N); ` `    ``return` `0; ` `} `

## Java

 `// Java program to find the number ` `// of ways to distribute N item ` `// among three people such ` `// that one person always gets ` `// the maximum value ` `class` `GFG{ ` ` `  `// Function to find the number ` `// of ways to distribute N ` `// items among 3 people ` `static` `int` `countWays(``int` `N) ` `{ ` `    ``// No distribution ` `    ``// possible ` `    ``if` `(N < ``4``) ` `        ``return` `0``; ` ` `  `    ``// Total number of ways to ` `    ``// distribute N items ` `    ``// among 3 people ` `    ``int` `ans = ((N - ``1``) * (N - ``2``)) / ``2``; ` ` `  `    ``// Store the number of ` `    ``// distributions which ` `    ``// are not possible ` `    ``int` `s = ``0``; ` ` `  `    ``for` `(``int` `i = ``2``; i <= N - ``3``; i++)  ` `    ``{ ` `        ``for` `(``int` `j = ``1``; j < i; j++) ` `        ``{ ` ` `  `            ``// Count possiblities ` `            ``// of two persons ` `            ``// receiving the ` `            ``// maximum ` `            ``if` `(N == ``2` `* i + j) ` `                ``s++; ` `        ``} ` `    ``} ` ` `  `    ``// If N is divisible by 3 ` `    ``if` `(N % ``3` `== ``0``) ` `        ``s = ``3` `* s + ``1``; ` `    ``else` `        ``s = ``3` `* s; ` ` `  `    ``// Return the final ` `    ``// count of ways ` `    ``// to distribute ` `    ``return` `ans - s; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `N = ``10``; ` `    ``System.out.println(countWays(N)); ` `} ` `} ` ` `  `// This code is contributed by rock_cool `

## Python3

 `# Python3 program to find the number ` `# of ways to distribute N item ` `# among three people such ` `# that one person always gets ` `# the maximum value ` ` `  `# Function to find the number ` `# of ways to distribute N ` `# items among 3 people ` `def` `countWays(N): ` `     `  `    ``# No distribution ` `    ``# possible ` `    ``if` `(N < ``4``): ` `        ``return` `0` ` `  `    ``# Total number of ways to ` `    ``# distribute N items ` `    ``# among 3 people ` `    ``ans ``=` `((N ``-` `1``) ``*` `(N ``-` `2``)) ``/``/` `2` ` `  `    ``# Store the number of ` `    ``# distributions which ` `    ``# are not possible ` `    ``s ``=` `0` ` `  `    ``for` `i ``in` `range``( ``2``, N ``-` `2``, ``1``): ` `        ``for` `j ``in` `range``( ``1``, i, ``1``): ` `             `  `            ``# Count possiblities ` `            ``# of two persons ` `            ``# receiving the ` `            ``# maximum ` `            ``if` `(N ``=``=` `2` `*` `i ``+` `j): ` `                ``s ``+``=` `1` ` `  `    ``# If N is divisible by 3 ` `    ``if` `(N ``%` `3` `=``=` `0``): ` `        ``s ``=` `3` `*` `s ``+` `1` ` `  `    ``else``: ` `        ``s ``=` `3` `*` `s ` ` `  `    ``# Return the final ` `    ``# count of ways ` `    ``# to distribute ` `    ``return` `ans ``-` `s ` ` `  `# Driver Code ` `N ``=` `10` ` `  `print` `(countWays(N)) ` `     `  `# This code is contributed by sanjoy_62 `

## C#

 `// C# program to find the number ` `// of ways to distribute N item ` `// among three people such ` `// that one person always gets ` `// the maximum value ` `using` `System; ` `class` `GFG{ ` ` `  `// Function to find the number ` `// of ways to distribute N ` `// items among 3 people ` `static` `int` `countWays(``int` `N) ` `{ ` `    ``// No distribution ` `    ``// possible ` `    ``if` `(N < 4) ` `        ``return` `0; ` ` `  `    ``// Total number of ways to ` `    ``// distribute N items ` `    ``// among 3 people ` `    ``int` `ans = ((N - 1) * (N - 2)) / 2; ` ` `  `    ``// Store the number of ` `    ``// distributions which ` `    ``// are not possible ` `    ``int` `s = 0; ` ` `  `    ``for` `(``int` `i = 2; i <= N - 3; i++)  ` `    ``{ ` `        ``for` `(``int` `j = 1; j < i; j++) ` `        ``{ ` ` `  `            ``// Count possiblities ` `            ``// of two persons ` `            ``// receiving the ` `            ``// maximum ` `            ``if` `(N == 2 * i + j) ` `                ``s++; ` `        ``} ` `    ``} ` ` `  `    ``// If N is divisible by 3 ` `    ``if` `(N % 3 == 0) ` `        ``s = 3 * s + 1; ` `    ``else` `        ``s = 3 * s; ` ` `  `    ``// Return the final ` `    ``// count of ways ` `    ``// to distribute ` `    ``return` `ans - s; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `Main() ` `{ ` `    ``int` `N = 10; ` `    ``Console.Write(countWays(N)); ` `} ` `} ` ` `  `// This code is contributed by Code_Mech `

Output:

```33
```

Time complexity: O(N2)
Auxiliary Space: O(1) My Personal Notes arrow_drop_up Check out this Author's contributed articles.

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Improved By : rock_cool, Code_Mech, sanjoy_62