Count different numbers that can be generated such that there digits sum is equal to 'n'
Last Updated : 09 May, 2022
Given an positive integer n. Count the different numbers that can be generated using digits 1, 2, 3 and 4 such that digits sum is the number 'n'. Here digit '4' will be treated as '1'. For instance,
32 = 3 + 2 = 5
1341 = 1 + 3 + 1 + 1 = 6
441 = 1 + 1 + 1 = 3
Note: Answer the value in mod = 109+7
Input: 2 Output: 5 Explanation There are only '5' numbers that can be made: 11 = 1 + 1 = 2 14 = 1 + 1 = 2 41 = 1 + 1 = 2 44 = 1 + 1 = 2 2 = 2 Input: 3 Output: 13 Explanation There are only '13' numbers that can be made i.e., 111, 114, 141, 144, 411, 414, 441, 444, 12, 21, 42, 24, 3.
The approach is to use Dynamic programming. The problem is same as coin change and Ways to write n as sum of two or more positive integers problems. The only difference is that, instead of iterating up-to 'n', iterate only from 1 to 3 as according to question, only 1, 2, 3 and 4 digits are allowed. But since '4' can be replaced with '1' therefore iterate through 1, 2 and 3 and double the count of '1' for compensation of digit '4'.
C++ // C++ program to count ways to write // 'n' as sum of digits #include<iostream> using namespace std; // Function to count 'num' as sum of // digits(1, 2, 3, 4) int countWays(int num) { // Initialize dp[] array int dp[num+1]; const int MOD = 1e9 + 7; // Base case dp[1] = 2; for(int i = 2; i <= num; ++i) { // Initialize the current dp[] // array as '0' dp[i] = 0; for(int j = 1; j <= 3; ++j) { /* if i == j then there is only one way to write with element itself 'i' */ if(i - j == 0) dp[i] += 1; /* If j == 1, then there exist two ways, one from '1' and other from '4' */ else if (j == 1) dp[i] += dp[i-j] * 2; /* if i - j is positive then pick the element from 'i-j' element of dp[] array */ else if(i - j > 0) dp[i] += dp[i-j]; // Check for modulus if(dp[i] >= MOD) dp[i] %= MOD; } } // return the final answer return dp[num]; } // Driver code int main() { int n = 3; cout << countWays(n); return 0; } // Java program to count ways to // write 'n' as sum of digits import java.io.*; public class GFG { // Function to count 'num' as // sum of digits(1, 2, 3, 4) static int countWays(int num) { // Initialize dp[] array int []dp= new int[num + 1]; int MOD = (int)1E9 + 7; // Base case dp[1] = 2; for(int i = 2; i <= num; ++i) { // Initialize the current // dp[] array as '0' dp[i] = 0; for(int j = 1; j <= 3; ++j) { // if i == j then there is // only one way to write with // element itself 'i' if(i - j == 0) dp[i] += 1; // If j == 1, then there exist // two ways, one from '1' and // other from '4' else if (j == 1) dp[i] += dp[i - j] * 2; // if i - j is positive then // pick the element from 'i-j' // element of dp[] array else if(i - j > 0) dp[i] += dp[i - j]; // Check for modulus if(dp[i] >= MOD) dp[i] %= MOD; } } // return the final answer return dp[num]; } // Driver code static public void main (String[] args) { int n = 3; System.out.println(countWays(n)); } } // This code is contributed by vt_m # Python3 program to count ways to write # 'n' as sum of digits # Function to count 'num' as sum of # digits(1, 2, 3, 4) def countWays(num): # Initialize dp[] array dp = [0] * (num + 1); MOD = 100000000 + 7; # Base case dp[1] = 2; for i in range(2, num + 1): # Initialize the current dp[] # array as '0' dp[i] = 0; for j in range(1, 4): # if i == j then there is only # one way to write with element # itself 'i' if(i - j == 0): dp[i] += 1; # If j == 1, then there exist # two ways, one from '1' and # other from '4' elif (j == 1): dp[i] += dp[i - j] * 2; # if i - j is positive then # pick the element from 'i-j' # element of dp[] array elif(i - j > 0): dp[i] += dp[i - j]; # Check for modulus if(dp[i] >= MOD): dp[i] %= MOD; # return the final answer return dp[num]; # Driver code n = 3; print(countWays(n)); # This code is contributed by mits
// C# program to count ways to // write 'n' as sum of digits using System; public class GFG { // Function to count 'num' as // sum of digits(1, 2, 3, 4) static int countWays(int num) { // Initialize dp[] array int []dp= new int[num + 1]; int MOD = (int)1E9 + 7; // Base case dp[1] = 2; for(int i = 2; i <= num; ++i) { // Initialize the current // dp[] array as '0' dp[i] = 0; for(int j = 1; j <= 3; ++j) { // if i == j then there is // only one way to write with // element itself 'i' if(i - j == 0) dp[i] += 1; // If j == 1, then there exist // two ways, one from '1' and // other from '4' else if (j == 1) dp[i] += dp[i - j] * 2; // if i - j is positive then // pick the element from 'i-j' // element of dp[] array else if(i - j > 0) dp[i] += dp[i - j]; // Check for modulus if(dp[i] >= MOD) dp[i] %= MOD; } } // return the final answer return dp[num]; } // Driver code static public void Main (String []args) { int n = 3; Console.WriteLine(countWays(n)); } } // This code is contributed by vt_m <?php // PHP program to count ways to write // 'n' as sum of digits // Function to count 'num' as sum of // digits(1, 2, 3, 4) function countWays($num) { // Initialize dp[] array $dp[$num + 1] = array(); $MOD = 100000000 + 7; // Base case $dp[1] = 2; for($i = 2; $i <= $num; ++$i) { // Initialize the current dp[] // array as '0' $dp[$i] = 0; for($j = 1; $j <= 3; ++$j) { /* if i == j then there is only one way to write with element itself 'i' */ if($i - $j == 0) $dp[$i] += 1; /* If j == 1, then there exist two ways, one from '1' and other from '4' */ else if ($j == 1) $dp[$i] += $dp[$i - $j] * 2; /* if i - j is positive then pick the element from 'i-j' element of dp[] array */ else if($i - $j > 0) $dp[$i] += $dp[$i - $j]; // Check for modulus if($dp[$i] >= $MOD) $dp[$i] %= $MOD; } } // return the final answer return $dp[$num]; } // Driver code $n = 3; echo countWays($n); // This code is contributed by jit_t ?> <script> // JavaScript program to count ways to // write 'n' as sum of digits // Function to count 'num' as // sum of digits(1, 2, 3, 4) function countWays(num) { // Initialize dp[] array let dp = []; let MOD = 1E9 + 7; // Base case dp[1] = 2; for(let i = 2; i <= num; ++i) { // Initialize the current // dp[] array as '0' dp[i] = 0; for(let j = 1; j <= 3; ++j) { // If i == j then there is // only one way to write with // element itself 'i' if (i - j == 0) dp[i] += 1; // If j == 1, then there exist // two ways, one from '1' and // other from '4' else if (j == 1) dp[i] += dp[i - j] * 2; // If i - j is positive then // pick the element from 'i-j' // element of dp[] array else if (i - j > 0) dp[i] += dp[i - j]; // Check for modulus if (dp[i] >= MOD) dp[i] %= MOD; } } // Return the final answer return dp[num]; } // Driver Code let n = 3; document.write(countWays(n)); // This code is contributed by susmitakundugoaldanga </script> Output
13
Time complexity: O(n)
Auxiliary space: O(n)
Note: Asked in Directi coding round(2014 and 2017)
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