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// 3343. Count Number of Balanced Permutations
#include "leetcode.h"
/*
* you are given a string 'num'. a string of digits is called balanced if the
* sum of the digits at even indices is equal to the sum of the digits at odd
* indices. return the number of distinct permutations of 'num' that are
* balanced. since the answer may be very large, return it modulo 10e9+7. a
* permutation is a rearrangement of all the characters of a string.
*/
static int mod = 1e9 + 7;
long long mod_exp(long long base, long long exp) {
long long res = 1;
base %= mod;
while (exp) {
if (exp & 1)
res = (res * base) % mod;
base = (base * base) % mod;
exp >>= 1;
}
return res;
}
int countBalancedPermutations(char *num) {
int n = strlen(num), freq[10] = {0};
for (int i = 0; i < n; i++)
freq[num[i] - '0']++;
int even_cnt = (n + 1) / 2, odd_cnt = n - even_cnt, sum = 0;
for (int i = 0; i < 10; i++)
sum += i * freq[i];
if (sum & 1)
return 0;
int target = sum / 2, max = n;
long long *fac = (long long *)malloc((max + 1) * sizeof(long long));
long long *ifac = (long long *)malloc((max + 1) * sizeof(long long));
fac[0] = 1;
for (int i = 1; i <= max; i++)
fac[i] = (fac[i - 1] * i) % mod;
ifac[max] = mod_exp(fac[max], mod - 2);
for (int i = max - 1; i >= 0; i--)
ifac[i] = (ifac[i + 1] * (i + 1)) % mod;
long long dp[11][42][730] = {{{0}}};
dp[0][0][0] = 1;
for (int i = 0; i < 10; i++) {
for (int j = 0; j <= even_cnt; j++) {
for (int k = 0; k <= target; k++) {
if (!dp[i][j][k])
continue;
int curr = freq[i];
for (int l = 0; l <= curr; l++) {
if (j + l > even_cnt)
break;
int ns = k + i * l;
if (ns > target)
break;
long long coef = (ifac[l] * ifac[curr - l]) % mod;
dp[i + 1][j + l][ns] =
(dp[i + 1][j + l][ns] + dp[i][j][k] * coef) % mod;
}
}
}
}
long long ans = dp[10][even_cnt][target];
ans = (ans * fac[even_cnt]) % mod;
ans = (ans * fac[odd_cnt]) % mod;
free(fac);
free(ifac);
return (int)ans;
}
int main() {
char *n1 = "123", *n2 = "112", *n3 = "12345";
printf("%d\n", countBalancedPermutations(n1)); // expect: 2
printf("%d\n", countBalancedPermutations(n2)); // expect: 1
printf("%d\n", countBalancedPermutations(n3)); // expect: 0
}
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