class Solution {
    /** Recursion */
    public int count(int coins[], int sum) {
        int n = coins.length;
        
        return countCombinations(coins, sum, n);
    }
    
    public int countCombinations(int[] coins, int sum, int n)
    {
        // base condition: when sum becomes 0 then we found valid combination
        if(sum == 0)
            return 1;
        
        // base condition: when we reach end of array, no valid combination found
        if(n == 0)
            return 0;
        
        if(coins[n-1] <= sum)
        {
            // include the current coin into combination, and this coin can be used later too
            int pick = countCombinations(coins, sum - coins[n-1], n);
            // do not include the current coin into combination
            int notPick = countCombinations(coins, sum, n-1);
            return pick + notPick;
        }
        // when current coin has value greater than the required sum, skip the coin
        return countCombinations(coins, sum, n-1);
    }
    
    
    /** Memoization */
    public int count(int coins[], int sum) {
        int n = coins.length;
        
        int[][] t = new int[n+1][sum+1];
        
        // when array size keeps on increasing but sum is 0, so we can have empty list
        for(int i=0; i<n+1; i++)
        {
            Arrays.fill(t[i], 1);
        }
        
        // when array size is 0, but sum keeps on increasing so we cannot have combination
        for(int j=0; j<n+1; j++)
        {
            Arrays.fill(t[j], 0);
        }
        
        return countCombinations(coins, sum, n, t);
    }
    
    public int countCombinations(int[] coins, int sum, int n, int[][] t)
    {
        // base condition: when sum becomes 0 then we found valid combination
        if(sum == 0)
            return 1;
        
        // base condition: when we reach end of array, no valid combination found
        if(n == 0)
            return 0;
        
        if(t[n][sum] != 0)
            return t[n][sum];
        
        if(coins[n-1] <= sum)
        {
            // include the current coin into combination, and this coin can be used later too
            int pick = countCombinations(coins, sum - coins[n-1], n, t);
            // do not include the current coin into combination
            int notPick = countCombinations(coins, sum, n-1, t);
            t[n][sum] = pick + notPick;
            return t[n][sum];
        }
        // when current coin has value greater than the required sum, skip the coin
        t[n][sum] = countCombinations(coins, sum, n-1, t);
        return t[n][sum];
    }
    
    /** Top-down approach */
    public int count(int coins[], int sum) {
        int n = coins.length;
        
        int[][] t = new int[n+1][sum+1];
        
        for(int i=0; i<n+1; i++)
        {
            for(int j=0; j<sum+1; j++)
            {
                // when array is empty
                if(i == 0)
                    t[i][j] = 0;
                
                // when sum is 0
                if(j == 0)
                    t[i][j] = 1;
            }
        }
        
        for(int i=1; i<n+1; i++)
        {
            for(int j=1; j<sum+1; j++)
            {
                if(coins[i-1] <= j)
                {
                    // include the current coin into combination, and this coin can be used later too
                    int pick = t[i][j - coins[i-1]];
                    // do not include the current coin into combination
                    int notPick = t[i-1][j];
                    t[i][j] = pick + notPick;
                }
                else
                    t[i][j] = t[i-1][j];
            }
        }
        return t[n][sum];
    }
}