# Count the number of ways to give ranks for N students such that same ranks are possible

• Difficulty Level : Expert
• Last Updated : 07 May, 2021

Given a number N which represents the number of students, the task is to calculate all possible ways to rank them according to their CGPA/marks, considering that two or more students can have the same rank. Since the answer can be large, perform the modulo with 109 + 7.

Examples:

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.  To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

In case you wish to attend live classes with experts, please refer DSA Live Classes for Working Professionals and Competitive Programming Live for Students.

Input: N = 1
Output:
Explanation:
There is only one way to rank a student, irrespective of his marks.

Input: N = 2
Output:
Explanation:
In the following two ways the ranks can be distributed among two students:
The high scoring student can be awarded the first rank and the low scoring student the second rank or both of them can be awarded the same rank if they have equal scores.

Approach: The idea for this problem is to use the Bell Numbers.

• A bell number is a number that counts the possible partitions of a set. Therefore, an N-th bell number is a number of non-empty subsets a set of size N can be partitioned into.
• For example, let’s consider the set {1, 2, 3} for N = 3. The bell number corresponding to N = 3 is 5. This signifies that the given set can be partitioned into following 5 non-empty subsets:
```{{1}, {2}, {3}}
{{1, 2}, {3}}
{{1, 3}, {2}}
{{2, 3}, {1}}
{{1, 2, 3}}```
• Clearly, the above bell numbers signify all the possible ranks. However, they do not calculate the permutations of the subset.
• Therefore, by multiplying every subset with K!, where K denotes the size of the respective subset, we get all the possible arrangements.
• As the same subproblems can be repeated, we can store the values of each subproblem in a data structure to optimize the complexity.

Below is the implementation of the above approach:

## C++

 `// C++ program to calculate the number``// of ways to give ranks for N``// students such that same ranks``// are possible` `#include ``using` `namespace` `std;` `const` `int` `mod = 1e9 + 7;` `// Initializing a table in order to``// store the bell triangle``vector > dp;` `// Function to calculate the K-th``// bell number``int` `f(``int` `n, ``int` `k)``{``    ``// If we have already calculated``    ``// the bell numbers until the``    ``// required N``    ``if` `(n < k)``        ``return` `0;` `    ``// Base case``    ``if` `(n == k)``        ``return` `1;` `    ``// First Bell Number``    ``if` `(k == 1)``        ``return` `1;` `    ``// If the value of the bell``    ``// triangle has already been``    ``// calculated``    ``if` `(dp[n][k] != -1)``        ``return` `dp[n][k];` `    ``// Fill the defined dp table``    ``return` `dp[n][k] = ((k * f(n - 1, k)) % mod``                       ``+ (f(n - 1, k - 1)) % mod)``                      ``% mod;``}` `// Function to return the number``// of ways to give ranks for N``// students such that same ranks``// are possible``long` `operation(``int` `n)``{``    ``// Resizing the dp table for the``    ``// given value of n``    ``dp.resize(n + 1, vector<``int``>(n + 1, -1));` `    ``// Variables to store the answer``    ``// and the factorial value``    ``long` `ans = 0, fac = 1;` `    ``// Iterating till N``    ``for` `(``int` `k = 1; k <= n; k++) {` `        ``// Simultaneously calculate the k!``        ``fac *= k;` `        ``// Computing the K-th bell number``        ``// and multiplying it with K!``        ``ans = (ans + (fac * f(n, k)) % mod)``              ``% mod;``    ``}``    ``return` `ans;``}` `// Driver code``int` `main()``{``    ``int` `n = 5;` `    ``cout << operation(n) << endl;` `    ``return` `0;``}`

## Java

 `// Java program to calculate the number``// of ways to give ranks for N``// students such that same ranks``// are possible``import` `java.util.*;` `class` `GFG{` `static` `int` `mod = (``int``)(1e9 + ``7``);` `// Initializing a table in order to``// store the bell triangle``static` `int` `[][]dp;` `// Function to calculate the K-th``// bell number``static` `int` `f(``int` `n, ``int` `k)``{``    ` `    ``// If we have already calculated``    ``// the bell numbers until the``    ``// required N``    ``if` `(n < k)``        ``return` `0``;` `    ``// Base case``    ``if` `(n == k)``        ``return` `1``;` `    ``// First Bell Number``    ``if` `(k == ``1``)``        ``return` `1``;` `    ``// If the value of the bell``    ``// triangle has already been``    ``// calculated``    ``if` `(dp[n][k] != -``1``)``        ``return` `dp[n][k];` `    ``// Fill the defined dp table``    ``return` `dp[n][k] = ((k * f(n - ``1``, k)) % mod +``                       ``(f(n - ``1``, k - ``1``)) % mod) % mod;``}` `// Function to return the number``// of ways to give ranks for N``// students such that same ranks``// are possible``static` `long` `operation(``int` `n)``{``    ` `    ``// Resizing the dp table for the``    ``// given value of n``    ``dp = ``new` `int``[n + ``1``][n + ``1``];``    ``for``(``int` `i = ``0``; i < n + ``1``; i++)``    ``{``       ``for``(``int` `j = ``0``; j < n + ``1``; j++)``       ``{``          ``dp[i][j] = -``1``;``       ``}``    ``}``    ` `    ``// Variables to store the answer``    ``// and the factorial value``    ``long` `ans = ``0``, fac = ``1``;` `    ``// Iterating till N``    ``for``(``int` `k = ``1``; k <= n; k++)``    ``{``       ` `       ``// Simultaneously calculate the k!``       ``fac *= k;``       ` `       ``// Computing the K-th bell number``       ``// and multiplying it with K!``       ``ans = (ans + (fac * f(n, k)) % mod) % mod;``    ``}``    ``return` `ans;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `n = ``5``;` `    ``System.out.print(operation(n) + ``"\n"``);``}``}` `// This code is contributed by amal kumar choubey`

## Python3

 `# Python3 program to calculate the number``# of ways to give ranks for N``# students such that same ranks``# are possible``mod ``=` `1e9` `+` `7` `# Initializing a table in order to``# store the bell triangle``dp ``=` `[[``-``1` `for` `x ``in` `range``(``6``)]``          ``for` `y ``in` `range``(``6``)]` `# Function to calculate the K-th``# bell number``def` `f(n, k):``    ` `    ``# If we have already calculated``    ``# the bell numbers until the``    ``# required N``    ``if` `(n < k):``        ``return` `0` `    ``# Base case``    ``if` `(n ``=``=` `k):``        ``return` `1` `    ``# First Bell Number``    ``if` `(k ``=``=` `1``):``        ``return` `1` `    ``# If the value of the bell``    ``# triangle has already been``    ``# calculated``    ``if` `(dp[n][k] !``=` `-``1``):``        ``return` `dp[n][k]` `    ``# Fill the defined dp table``    ``dp[n][k] ``=` `((((k ``*` `f(n ``-` `1``, k)) ``%` `mod ``+``               ``(f(n ``-` `1``, k ``-` `1``)) ``%` `mod) ``%` `mod))``    ``return` `dp[n][k]` `# Function to return the number``# of ways to give ranks for N``# students such that same ranks``# are possible``def` `operation(n):` `    ``# Resizing the dp table for the``    ``# given value of n``    ``global` `dp` `    ``# Variables to store the answer``    ``# and the factorial value``    ``ans ``=` `0``    ``fac ``=` `1` `    ``# Iterating till N``    ``for` `k ``in` `range``(``1``, n ``+` `1``):` `        ``# Simultaneously calculate the k!``        ``fac ``*``=` `k` `        ``# Computing the K-th bell number``        ``# and multiplying it with K!``        ``ans ``=` `(ans ``+` `(fac ``*` `f(n, k)) ``%` `mod) ``%` `mod``        ` `    ``return` `ans` `# Driver code``if` `__name__ ``=``=` `"__main__"``:` `    ``n ``=` `5` `    ``print``(``int``(operation(n)))` `# This code is contributed by ukasp`

## C#

 `// C# program to calculate the number``// of ways to give ranks for N``// students such that same ranks``// are possible``using` `System;` `class` `GFG{` `static` `int` `mod = (``int``)(1e9 + 7);` `// Initializing a table in order to``// store the bell triangle``static` `int` `[,]dp;` `// Function to calculate the K-th``// bell number``static` `int` `f(``int` `n, ``int` `k)``{``    ` `    ``// If we have already calculated``    ``// the bell numbers until the``    ``// required N``    ``if` `(n < k)``        ``return` `0;` `    ``// Base case``    ``if` `(n == k)``        ``return` `1;` `    ``// First Bell Number``    ``if` `(k == 1)``        ``return` `1;` `    ``// If the value of the bell``    ``// triangle has already been``    ``// calculated``    ``if` `(dp[n, k] != -1)``        ``return` `dp[n, k];` `    ``// Fill the defined dp table``    ``return` `dp[n, k] = ((k * f(n - 1, k)) % mod +``                       ``(f(n - 1, k - 1)) % mod) % mod;``}` `// Function to return the number``// of ways to give ranks for N``// students such that same ranks``// are possible``static` `long` `operation(``int` `n)``{``    ` `    ``// Resizing the dp table for the``    ``// given value of n``    ``dp = ``new` `int``[n + 1, n + 1];``    ``for``(``int` `i = 0; i < n + 1; i++)``    ``{``       ``for``(``int` `j = 0; j < n + 1; j++)``       ``{``          ``dp[i, j] = -1;``       ``}``    ``}``    ` `    ``// Variables to store the answer``    ``// and the factorial value``    ``long` `ans = 0, fac = 1;` `    ``// Iterating till N``    ``for``(``int` `k = 1; k <= n; k++)``    ``{``       ` `       ``// Simultaneously calculate the k!``       ``fac *= k;``       ` `       ``// Computing the K-th bell number``       ``// and multiplying it with K!``       ``ans = (ans + (fac * f(n, k)) % mod) % mod;``    ``}``    ``return` `ans;``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``int` `n = 5;` `    ``Console.Write(operation(n) + ``"\n"``);``}``}` `// This code is contributed by amal kumar choubey`

## Javascript

 ``
Output:
`541`

My Personal Notes arrow_drop_up