coinchange.cpp

#include<iostream>
using namespace std;

int dp[1001][1001]; // since constraint is given of 1000, we take dp[1001][1001]
int numberOfWays(int coins[], int n, int sum){
    for(int i = 0; i<n+1; i++){
        for(int j = 0; j<sum+1; j++){
            if(i == 0){ // if coin's value is 0, we can't form the sum
                dp[i][j] = false;
            }
            if(j == 0){ // if sum is 0, there will always be one way to form sum 0
                dp[i][j] = true;
            }
        }
    }
    for(int i = 1; i<n+1; i++){
        for(int j = 1; j<sum+1; j++){
            if(coins[i-1]<=j){ // if the coin's value is less than the given sum
                // there will be two possibilities whether to take it or leave it
                // Total    // we pick it so sum got reduced and we will stay on the same index     
                dp[i][j] = dp[i][j-coins[i-1]] + dp[i-1][j]; 
                                                 // we didn't pick it
            }
            else{
                dp[i][j] = dp[i-1][j]; // if we didn't pick it then the sum remains same and we will move to the next index
            }
        }
    }
    return dp[n][sum]; // return ans
}

int main(){
    int n, sum;
    cin>>n>>sum;

    int coins[n];
    for(int i = 0; i<n; i++){
        cin>>coins[i];
    }
    cout<<numberOfWays(coins,n,sum);

    return 0;
}