# Number of compositions of a natural number

Given a natural number n, find the number of ways in which n can be expressed as a sum of natural numbers when order is taken into consideration. Two sequences that differ in the order of their terms define different compositions of their sum.

Examples:

Input : 4 Output : 8 Explanation All 8 position composition are: 4, 1+3, 3+1, 2+2, 1+1+2, 1+2+1, 2+1+1 and 1+1+1+1 Input : 8 Output : 128

A Simple Solution is to generate all compositions and count them.

Using the concept of combinatorics, it can be proved that any natural number n will have **2^(n-1) distinct compositions** when order is taken into consideration.

One way to see why the answer is 2^(n-1) directly is to write n as a sum of 1s:

n = 1 + 1 + 1 +…+ 1 (n times).There are (n-1) plus signs between all 1s. For every plus sign we can choose to split ( by putting a bracket) at the point or not split. Therefore answer is 2^(n-1).

For example, n = 4

No Split

4 = 1 + 1 + 1 + 1 [We write as single 4]Different ways to split once

4 = (1) + (1 + 1 + 1) [We write as 1 + 3]

4 = (1 + 1) + (1 + 1) [We write as 2 + 2]

4 = (1 + 1 + 1) + (1) [We write as 3 + 1]Different ways to split twice

4 = (1) + (1 + 1) + (1) [We write as 1 + 2 + 1]

4 = (1 + 1) + (1) + (1) [We write as 2 + 1 + 1]

4 = (1) + (1) + (1 + 1) [We write as 1 + 1 + 2]Different ways to split three times

4 = (1) + (1) + (1) + (1) [We write as 1 + 1 + 1 + 1]Since there are

(n-1)plus signs between the n 1s, there are 2^(n-1) ways of choosing where to split the sum, and hence 2^(n-1) possible sums .

## C++

`// C++ program to find the total number of ` `// compositions of a natural number ` `#include<iostream> ` `using` `namespace` `std; ` ` ` `#define ull unsigned long long ` ` ` `ull countCompositions(ull n) ` `{ ` ` ` `// Return 2 raised to power (n-1) ` ` ` `return` `(1L) << (n-1); ` `} ` ` ` `// Driver Code ` `int` `main() ` `{ ` ` ` `ull n = 4; ` ` ` `cout << countCompositions(n) << ` `"\n"` `; ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to find ` `// the total number of ` `// compositions of a ` `// natural number ` `import` `java.io.*; ` `import` `java.util.*; ` ` ` `class` `GFG ` `{ ` `public` `static` `int` `countCompositions(` `int` `n) ` `{ ` ` ` `// Return 2 raised ` ` ` `// to power (n-1) ` ` ` `return` `1` `<< (n - ` `1` `); ` `} ` ` ` `// Driver Code ` `public` `static` `void` `main(String args[]) ` `{ ` ` ` `int` `n = ` `4` `; ` ` ` `System.out.print(countCompositions(n)); ` `} ` `} ` ` ` `// This code is contributed by ` `// Akanksha Rai(Abby_akku) ` |

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## Python

`# Python code to find the total number of ` `# compositions of a natural number ` `def` `countCompositions(n): ` ` ` ` ` `# function to return the total number ` ` ` `# of composition of n ` ` ` `return` `(` `2` `*` `*` `(n` `-` `1` `)) ` ` ` `# Driver Code ` `print` `(countCompositions(` `4` `)) ` |

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## C#

`// C# program to find the ` `// total number of compositions ` `// of a natural number ` `using` `System; ` ` ` `class` `GFG ` `{ ` `public` `static` `int` `countCompositions(` `int` `n) ` `{ ` ` ` `// Return 2 raised ` ` ` `// to power (n-1) ` ` ` `return` `1 << (n - 1); ` `} ` ` ` `// Driver Code ` `public` `static` `void` `Main() ` `{ ` ` ` `int` `n = 4; ` ` ` `Console.Write(countCompositions(n)); ` `} ` `} ` ` ` `// This code is contributed by mits ` |

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## PHP

`<?php ` `// PHP program to find the ` `// total number of compositions ` `// of a natural number ` ` ` `function` `countCompositions(` `$n` `) ` `{ ` ` ` `// Return 2 raised ` ` ` `// to power (n-1) ` ` ` `return` `((1) << (` `$n` `- 1)); ` `} ` ` ` `// Driver Code ` `$n` `= 4; ` `echo` `countCompositions(` `$n` `), ` `"\n"` `; ` ` ` `// This code is contributed ` `// by ajit ` `?> ` |

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**Output:**

8

This article is contributed by **Sruti Rai **. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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