# Problem Statement

The purpose of this blog is to explain the problem of achieving a final score, given the set of input balls. Let's understand a pool game first. A pool table consists of six pockets and balls numbered from 1 to 15. A player scores when he pockets a ball (taking the ball numbered two gives the equivalent two).
Let's take an example of a custom pool game with valid balls, which is the selection of only a few numbered balls from the set. We have to find the number of ways a player can reach a final score using valid balls.
Let's say that the user has selected three balls (2, 4 and 6), and played the pool table, and his final score is 6. There are three number of ways the player can score six by pocketing [{2,2,2}, {2,4}, {6}].
We have to define a method that accepts valid and final input and return number of ways to reach a score.

# Explanation

We have an input array named valid to store the chosen balls and a number final defining the final score of the player.
We define an array named ways to store the number of ways to reach an element denoted by index. Example, ways[i] means the number of ways to reach i.
Let's say that the user has selected three balls (2, 4 and 6) and achieved a final score of 6.
Given,

``````valid = [2, 4, 6]
final = 6
ways = [0, 0, 0, 0, 0, 0]
``````

The idea is to iterate over each valid score and try to find how it impacts on the number of ways to achieve a higher score.
First Iteration for Valid Score (2):
Set the number of ways to reach 2, denoted by ways

``````ways = ways + 1
ways = 1
``````

We know that two is a valid score means we can find ways to reach a final score of 3, using ways to reach a final score of 1

``````ways = ways + ways
ways = 0
``````

similarly

``````ways = ways + ways
ways = 1
ways = ways + ways
ways = 0
ways = ways + ways
ways = 1
ways = [0, 1, 0, 1, 0, 1]
``````

Second Iteration for Valid Score (4):
Set the number of ways to reach 4, denoted by ways

``````ways = ways + 1
ways = 2
``````

We know that four is a valid score means we can find ways to reach a final score of 5, using ways to reach a final score of 1

``````ways = ways + ways
ways = 0
``````

similarly

``````ways = ways + ways
ways = 2
ways = [0, 1, 0, 2, 0, 2]
``````

Third Iteration for Valid Score (6):
Set the number of ways to reach 6, denoted by ways

``````ways = ways + 1
ways = 3
ways = [0, 1, 0, 2, 0, 3]
``````

We can get the number of ways to reach six, as three (using ways).

# Code

``````//Rextester.Program.Main is the entry point for your code. Don't change it.
//Compiler version 4.0.30319.17929 for Microsoft (R) .NET Framework 4.5

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text.RegularExpressions;

namespace Rextester
{
public class Program
{
public static void Main(string[] args)
{
// Input
// Valid Points - 2, 4, 6
// Final Point - 6

// Output - Ways to achieve final point using valid points without specific combination
// 6 can be achieved using 3 ways - [2,2,2], [2,4] and 
// 8 can be made using 4 ways - [2,2,2,2], [2,2,4], [2,6] and 
Console.WriteLine(NumberOfWays(new int[]{2, 4, 6}, 6));
Console.WriteLine(NumberOfWays(new int[]{2, 4, 6}, 8));
}

public static int NumberOfWays(int[] valid, int final) {
if(valid == null || valid.Length == 0 || final <= 0) {
return 0;
}
var ways = new int[final + 1];
// Base condition to denote a number can be formed by it's own
// e.g. 2 can be formed if 2 is a valid score and so on
ways = 1;

// The idea is to perform additive operation for all the valid scores
// until we reach the final
foreach(var input in valid) {
for(var i = input; i <= final; i++) {
ways[i] += ways[i-input];
}
}

return ways[final];
}
}
}``````