# Count ways to express a number as sum of powers

Given two integers x and n, we need to find number of ways to express x as sum of n-th powers of unique natural numbers. It is given that 1 <= n <= 20.

Examples:

Input : x = 100 n = 2 Output : 3 Explanation: There are three ways to express 100 as sum of natural numbers raised to power 2. 100 = 10^2 = 8^2+6^2 = 1^2+3^2+4^2+5^2+7^2 Input : x = 100 n = 3 Output : 1 Explanation : The only combination is, 1^3 + 2^3 + 3^3 + 4^3

We use recursion to solve the problem. We first check one by one that the number is included in summation or not.

## C++

`// C++ program to count number of ways ` `// to express x as sum of n-th power ` `// of unique natural numbers. ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// num is current num. ` `int` `countWaysUtil(` `int` `x, ` `int` `n, ` `int` `num) ` `{ ` ` ` `// Base cases ` ` ` `int` `val = (x - ` `pow` `(num, n)); ` ` ` `if` `(val == 0) ` ` ` `return` `1; ` ` ` `if` `(val < 0) ` ` ` `return` `0; ` ` ` ` ` `// Consider two possibilities, num is ` ` ` `// included and num is not included. ` ` ` `return` `countWaysUtil(val, n, num + 1) + ` ` ` `countWaysUtil(x, n, num + 1); ` `} ` ` ` `// Returns number of ways to express ` `// x as sum of n-th power of two. ` `int` `countWays(` `int` `x, ` `int` `n) ` `{ ` ` ` `return` `countWaysUtil(x, n, 1); ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `x = 100, n = 2; ` ` ` `cout << countWays(x, n); ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to count number of ways ` `// to express x as sum of n-th power ` `// of unique natural numbers. ` `public` `class` `GFG { ` ` ` ` ` `// num is current num. ` ` ` `static` `int` `countWaysUtil(` `int` `x, ` `int` `n, ` `int` `num) ` ` ` `{ ` ` ` `// Base cases ` ` ` `int` `val = (` `int` `) (x - Math.pow(num, n)); ` ` ` `if` `(val == ` `0` `) ` ` ` `return` `1` `; ` ` ` `if` `(val < ` `0` `) ` ` ` `return` `0` `; ` ` ` ` ` `// Consider two possibilities, num is ` ` ` `// included and num is not included. ` ` ` `return` `countWaysUtil(val, n, num + ` `1` `) + ` ` ` `countWaysUtil(x, n, num + ` `1` `); ` ` ` `} ` ` ` ` ` `// Returns number of ways to express ` ` ` `// x as sum of n-th power of two. ` ` ` `static` `int` `countWays(` `int` `x, ` `int` `n) ` ` ` `{ ` ` ` `return` `countWaysUtil(x, n, ` `1` `); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `main(String args[]) ` ` ` `{ ` ` ` `int` `x = ` `100` `, n = ` `2` `; ` ` ` `System.out.println(countWays(x, n)); ` ` ` `} ` `} ` `// This code is contributed by Sumit Ghosh ` |

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## Python3

`# Python program to count number of ways ` `# to express x as sum of n-th power ` `# of unique natural numbers. ` ` ` `# num is current num. ` `def` `countWaysUtil(x,n,num): ` ` ` ` ` `# Base cases ` ` ` `val ` `=` `(x ` `-` `pow` `(num, n)) ` ` ` `if` `(val ` `=` `=` `0` `): ` ` ` `return` `1` ` ` `if` `(val < ` `0` `): ` ` ` `return` `0` ` ` ` ` `# Consider two possibilities, num is ` ` ` `# included and num is not included. ` ` ` `return` `countWaysUtil(val, n, num ` `+` `1` `) ` `+` `\ ` ` ` `countWaysUtil(x, n, num ` `+` `1` `) ` ` ` ` ` `# Returns number of ways to express ` `# x as sum of n-th power of two. ` `def` `countWays(x,n): ` ` ` `return` `countWaysUtil(x, n, ` `1` `) ` ` ` ` ` `# Driver code ` `x ` `=` `100` `n ` `=` `2` ` ` `print` `(countWays(x, n)) ` ` ` `# This code is contributed ` `# by Anant Agarwal. ` |

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## C#

`// C# program to count number of ways ` `// to express x as sum of n-th power ` `// of unique natural numbers. ` `using` `System; ` ` ` `public` `class` `GFG { ` ` ` ` ` `// num is current num. ` ` ` `static` `int` `countWaysUtil(` `int` `x, ` ` ` `int` `n, ` `int` `num) ` ` ` `{ ` ` ` ` ` `// Base cases ` ` ` `int` `val = (` `int` `) (x - Math.Pow(num, n)); ` ` ` `if` `(val == 0) ` ` ` `return` `1; ` ` ` `if` `(val < 0) ` ` ` `return` `0; ` ` ` ` ` `// Consider two possibilities, ` ` ` `// num is included and num is ` ` ` `// not included. ` ` ` `return` `countWaysUtil(val, n, num + 1) ` ` ` `+ countWaysUtil(x, n, num + 1); ` ` ` `} ` ` ` ` ` `// Returns number of ways to express ` ` ` `// x as sum of n-th power of two. ` ` ` `static` `int` `countWays(` `int` `x, ` `int` `n) ` ` ` `{ ` ` ` `return` `countWaysUtil(x, n, 1); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `Main() ` ` ` `{ ` ` ` `int` `x = 100, n = 2; ` ` ` ` ` `Console.WriteLine(countWays(x, n)); ` ` ` `} ` `} ` ` ` `// This code is contributed by vt_m. ` |

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## PHP

`<?php ` `// PHP program to count number of ways ` `// to express x as sum of n-th power ` `// of unique natural numbers. ` ` ` `// num is current num. ` `function` `countWaysUtil(` `$x` `, ` `$n` `, ` `$num` `) ` `{ ` ` ` ` ` `// Base cases ` ` ` `$val` `= (` `$x` `- pow(` `$num` `, ` `$n` `)); ` ` ` `if` `(` `$val` `== 0) ` ` ` `return` `1; ` ` ` `if` `(` `$val` `< 0) ` ` ` `return` `0; ` ` ` ` ` `// Consider two possibilities, num is ` ` ` `// included and num is not included. ` ` ` `return` `(countWaysUtil(` `$val` `, ` `$n` `, ` `$num` `+ 1) + ` ` ` `countWaysUtil(` `$x` `, ` `$n` `, ` `$num` `+ 1)); ` `} ` ` ` `// Returns number of ways to express ` `// x as sum of n-th power of two. ` `function` `countWays(` `$x` `, ` `$n` `) ` `{ ` ` ` `return` `countWaysUtil(` `$x` `, ` `$n` `, 1); ` `} ` ` ` `// Driver code ` `$x` `= 100; ` `$n` `= 2; ` `echo` `(countWays(` `$x` `, ` `$n` `)); ` ` ` `// This code is contributed by Ajit. ` `?> ` |

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Output:

3

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