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Number of ways to obtain each numbers in range [1, b+c] by adding any two numbers in range [a, b] and [b, c]
• Difficulty Level : Expert
• Last Updated : 05 Aug, 2020

Given three integers a, b and c. You need to select one integer from the range [a, b] and one integer from the range [b, c] and add them. The task to calculate the number of ways to obtain the sum for all the numbers in the range [1, b+c].
Examples:

Input: a = 1, b = 2, c = 2
Output: 0, 0, 1, 1
Explanation:
The numbers to be obtained are [1, b+c] = [1, 4] = {1, 2, 3, 4}
Therefore, the number of ways to obtain each are:
1 – can’t be obtained
2 – can’t be obtained
3 – only one way. select {1} from range [a, b] and {2} from range [b, c] – 1 + 2 = 3
4 – only one way. select {2} from range [a, b] and {2} from range [b, c] – 2 + 2 = 4
Input: a = 1, b = 3, c = 4
Output: 0, 0, 0, 1, 2, 2, 1

Simple Approach:

• A simple brute force solution will be to use a nested loop where exterior loop traverses from i = a to i = b and inner loop from j = b to j = c inclusive.

• We will initialise array a of size b + c + 1 with zero. Now in loops, we will increment the index at i+j, i.e., (a[i+j]++)

• We will simply print the array at the end.

Below is the implementation of the above approach.

## C++

 `// C++ program to calculate` `// the number of ways`   `#include ` `using` `namespace` `std;`   `void` `CountWays(``int` `a, ``int` `b, ``int` `c)` `{` `    ``int` `x = b + c + 1;` `    ``int` `arr[x] = { 0 };`   `    ``// Initialising the array with zeros.` `    ``// You can do using memset too.` `    ``for` `(``int` `i = a; i <= b; i++) {` `        ``for` `(``int` `j = b; j <= c; j++) {` `            ``arr[i + j]++;` `        ``}` `    ``}` `    ``// Printing the array` `    ``for` `(``int` `i = 1; i < x; i++) {` `        ``cout << arr[i] << ``" "``;` `    ``}` `    ``cout << endl;` `}` `// Driver code` `int` `main()` `{` `    ``int` `a = 1;` `    ``int` `b = 2;` `    ``int` `c = 2;`   `    ``CountWays(a, b, c);`   `    ``return` `0;` `}`

## Java

 `// Java program to calculate ` `// the number of ways ` `class` `GFG{` `    `  `public` `static` `void` `CountWays(``int` `a, ``int` `b,` `                                    ``int` `c) ` `{ ` `    ``int` `x = b + c + ``1``; ` `    ``int``[] arr = ``new` `int``[x]; ` `    `  `    ``// Initialising the array with zeros. ` `    ``// You can do using memset too. ` `    ``for``(``int` `i = a; i <= b; i++)` `    ``{ ` `       ``for``(``int` `j = b; j <= c; j++)` `       ``{ ` `          ``arr[i + j]++; ` `       ``} ` `    ``} ` `    `  `    ``// Printing the array ` `    ``for``(``int` `i = ``1``; i < x; i++) ` `    ``{` `       ``System.out.print(arr[i] + ``" "``);` `    ``} ` `} `   `// Driver code` `public` `static` `void` `main(String[] args)` `{` `    ``int` `a = ``1``; ` `    ``int` `b = ``2``; ` `    ``int` `c = ``2``; ` `    `  `    ``CountWays(a, b, c); ` `}` `}`   `// This code is contributed by divyeshrabadiya07`

## Python3

 `# Python3 program to calculate` `# the number of ways` `def` `CountWays(a, b, c):` `    `  `    ``x ``=` `b ``+` `c ``+` `1``;` `    ``arr ``=` `[``0``] ``*` `x;`   `    ``# Initialising the array with zeros.` `    ``# You can do using memset too.` `    ``for` `i ``in` `range``(a, b ``+` `1``):` `        ``for` `j ``in` `range``(b, c ``+` `1``):` `            ``arr[i ``+` `j] ``+``=` `1``;`   `    ``# Printing the array` `    ``for` `i ``in` `range``(``1``, x):` `        ``print``(arr[i], end ``=` `" "``);` `        `  `# Driver code` `if` `__name__ ``=``=` `'__main__'``:` `    `  `    ``a ``=` `1``;` `    ``b ``=` `2``;` `    ``c ``=` `2``;`   `    ``CountWays(a, b, c);` `    `  `# This code is contributed by Rajput-Ji`

## C#

 `// C# program to calculate ` `// the number of ways ` `using` `System;` `class` `GFG{ ` `    `  `public` `static` `void` `CountWays(``int` `a, ``int` `b, ` `                                    ``int` `c) ` `{ ` `    ``int` `x = b + c + 1; ` `    ``int``[] arr = ``new` `int``[x]; ` `    `  `    ``// Initialising the array with zeros. ` `    ``// You can do using memset too. ` `    ``for``(``int` `i = a; i <= b; i++) ` `    ``{ ` `        ``for``(``int` `j = b; j <= c; j++) ` `        ``{ ` `            ``arr[i + j]++; ` `        ``} ` `    ``} ` `    `  `    ``// Printing the array ` `    ``for``(``int` `i = 1; i < x; i++) ` `    ``{ ` `        ``Console.Write(arr[i] + ``" "``); ` `    ``} ` `} `   `// Driver code ` `public` `static` `void` `Main() ` `{ ` `    ``int` `a = 1; ` `    ``int` `b = 2; ` `    ``int` `c = 2; ` `    `  `    ``CountWays(a, b, c); ` `} ` `} `   `// This code is contributed by rutvik_56`

Output:

```0 0 1 1

```

Time Complexity: O((b-a)*(c-b)), which in the worst case is O(c2)
Efficient Approach: The idea is to use Prefix Sum logic to solve this problem.

1. We will traverse i from [a, b] and for every i we will simply increment the value of starting interval arr[i + b] by 1 and decrement the value of ending interval arr[i + c + 1] by 1.

2. Now all we need to do is to calculate the prefix sum of the array ( arr[i]+ = arr[i-1] ) and print the array.

Lets see the approach with the help of an example.
Why does this work?

For example: a = 1, b = 2, c = 2, we will encounter only two values of i
i = 1 = > arr[1+2]++; arr[1+2+1]–;
i = 2 = > arr[2+2]++; arr[2+2+1]–;
arr = {0, 0, 0, 1, 0, -1};
prefix sums:
arr = {0, 0, 0, 1, 1, 0};
Now carefully look and realise that this is our answer.
So what we do at particular index i is arr[i+b]++ and arr[i+c+1]–;
Now we are using prefix sums so all the values will be incremented by 1 between i+b and infinite(We won’t go there but will result in prefix sum increment by 1 and as soon as we do a decrement at i+c+1 all the values between i+c+1 and infinite will be decreased by one.
So effectively all the values in the range [i+b, i+c] are incremented by one, and rest all the values will remain unaffected.

Below is the implementation of the above approach.

## C++

 `// C++ program to calculate` `// the number of ways`   `#include ` `using` `namespace` `std;`   `void` `CountWays(``int` `a, ``int` `b, ``int` `c)` `{` `    ``// 2 is added because sometimes` `    ``// we will decrease the` `    ``// value out of bounds.` `    ``int` `x = b + c + 2;`   `    ``// Initialising the array with zeros.` `    ``// You can do using memset too.` `    ``int` `arr[x] = { 0 };`   `    ``for` `(``int` `i = 1; i <= b; i++) {` `        ``arr[i + b]++;` `        ``arr[i + c + 1]--;` `    ``}`   `    ``// Printing the array` `    ``for` `(``int` `i = 1; i < x - 1; i++) {` `        ``arr[i] += arr[i - 1];` `        ``cout << arr[i] << ``" "``;` `    ``}` `    ``cout << endl;` `}`   `// Driver code` `int` `main()` `{` `    ``int` `a = 1;` `    ``int` `b = 2;` `    ``int` `c = 2;`   `    ``CountWays(a, b, c);`   `    ``return` `0;` `}`

## Java

 `// Java program to calculate` `// the number of ways` `import` `java.util.*;`   `class` `GFG{`   `static` `void` `CountWays(``int` `a, ``int` `b, ``int` `c)` `{` `    `  `    ``// 2 is added because sometimes` `    ``// we will decrease the` `    ``// value out of bounds.` `    ``int` `x = b + c + ``2``;`   `    ``// Initialising the array with zeros.` `    ``int` `arr[] = ``new` `int``[x];`   `    ``for``(``int` `i = ``1``; i <= b; i++)` `    ``{` `       ``arr[i + b]++;` `       ``arr[i + c + ``1``]--;` `    ``}`   `    ``// Printing the array` `    ``for``(``int` `i = ``1``; i < x - ``1``; i++)` `    ``{` `       ``arr[i] += arr[i - ``1``];` `       ``System.out.print(arr[i] + ``" "``);` `    ``}` `    ``System.out.println();` `}`   `// Driver code` `public` `static` `void` `main(String[] args)` `{` `    ``int` `a = ``1``;` `    ``int` `b = ``2``;` `    ``int` `c = ``2``;`   `    ``CountWays(a, b, c);` `}` `}`   `// This code is contributed by Rohit_ranjan`

## C#

 `// C# program to calculate` `// the number of ways` `using` `System;` `class` `GFG{`   `static` `void` `CountWays(``int` `a, ``int` `b, ``int` `c)` `{` `    `  `    ``// 2 is added because sometimes` `    ``// we will decrease the` `    ``// value out of bounds.` `    ``int` `x = b + c + 2;`   `    ``// Initialising the array with zeros.` `    ``int` `[]arr = ``new` `int``[x];`   `    ``for``(``int` `i = 1; i <= b; i++)` `    ``{` `        ``arr[i + b]++;` `        ``arr[i + c + 1]--;` `    ``}`   `    ``// Printing the array` `    ``for``(``int` `i = 1; i < x - 1; i++)` `    ``{` `        ``arr[i] += arr[i - 1];` `        ``Console.Write(arr[i] + ``" "``);` `    ``}` `    ``Console.WriteLine();` `}`   `// Driver code` `public` `static` `void` `Main()` `{` `    ``int` `a = 1;` `    ``int` `b = 2;` `    ``int` `c = 2;`   `    ``CountWays(a, b, c);` `}` `}`   `// This code is contributed by Code_Mech`

## Python3

 `# Python3 program to calculate` `# the number of ways` `def` `CountWays(a, b, c):` `     `  `    ``# 2 is added because sometimes` `    ``# we will decrease the` `    ``# value out of bounds.` `    ``x ``=` `b ``+` `c ``+` `2``;` ` `  `    ``# Initialising the array with zeros.` `    ``arr ``=` `[``0``] ``*` `x;` ` `  `    ``for` `i ``in` `range``(``1``, b``+``1``):` `       ``arr[i ``+` `b] ``=` `arr[i ``+` `b] ``+` `1``;` `       ``arr[i ``+` `c ``+` `1``] ``=` `arr[i ``+` `c ``+` `1``] ``-``1``;` `    `  ` `  `    ``# Printing the array` `    ``for` `i ``in` `range``(``1``, x``-``1``):` `    `  `       ``arr[i] ``+``=` `arr[i ``-` `1``];` `       ``print``(arr[i], end ``=` `" "``);`   ` `  `# Driver code` `if` `__name__ ``=``=` `'__main__'``:` `     `  `    ``a ``=` `1``;` `    ``b ``=` `2``;` `    ``c ``=` `2``;` ` `  `    ``CountWays(a, b, c);` `     `  `# This code is contributed by rock_cool`

Output:

```0 0 1 1

```

Time complexity: O(C)

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