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 = **10 ^{9}+7**

Input:2Output:5ExplanationThere 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 = 2Input:3Output:13ExplanationThere 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++ 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 modulas 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; }

Output13

**Time complexity: **O(n)

**Auxiliary space: **O(n)

**Note:** Asked in Directi coding round(2014 and 2017)

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