Language: C++

#include <bits/stdc++.h>
 
#define _overload(_1,_2,_3,name,...) name
#define _rep(i,n) _range(i,0,n)
#define _range(i,a,b) for(int i=(int)(a);i<(int)(b);++i)
#define rep(...) _overload(__VA_ARGS__,_range,_rep,)(__VA_ARGS__)
 
#define _rrep(i,n) _rrange(i,n,0)
#define _rrange(i,a,b) for(int i=(int)(a)-1;i>=(int)(b);--i)
#define rrep(...) _overload(__VA_ARGS__,_rrange,_rrep,)(__VA_ARGS__)
 
#define _all(arg) begin(arg),end(arg)
#define uniq(arg) sort(_all(arg)),(arg).erase(unique(_all(arg)),end(arg))
#define getidx(ary,key) lower_bound(_all(ary),key)-begin(ary)
#define clr(a,b) memset((a),(b),sizeof(a))
#define bit(n) (1LL<<(n))
 
// #define DEBUG
 
#ifdef DEBUG
    #define dump(...) fprintf(stderr, __VA_ARGS__)
#else
    #define dump(...)
#endif
 
template<class T>bool chmax(T &a, const T &b) { return (a<b)?(a=b,1):0;}
template<class T>bool chmin(T &a, const T &b) { return (b<a)?(a=b,1):0;}
 
using namespace std;
using ll=long long;
using vi=vector<int>;
using vll=vector<ll>;
 
const double EPS = 1e-10;
const double PI = acos(-1.0);
const ll inf =1LL << 62;
const ll mod=1000000007LL;
const int dx[4]={1,0,-1,0};
const int dy[4]={0,1,0,-1};
 
 
ll extgcd(ll a,ll b,ll& x,ll& y){x=1,y=0;ll g=a;if(b!=0) g=extgcd(b,a%b,y,x),y-=a/b*x;return g;}
ll ADD(const ll &a, const ll &b,const ll &mod) { return (a+b)%mod;}
ll SUB(const ll &a, const ll &b,const ll &mod) { return (a-b+mod)%mod;}
ll MUL(const ll &a, const ll &b,const ll &mod) { return (1LL*a*b)%mod;}
ll DIV(const ll &a, const ll &b,const ll &mod) {ll x,y; extgcd(b,mod,x,y);return MUL(a,(x+mod)%mod,mod);}
 
random_device rd;
mt19937 mt(rd());
uniform_int_distribution<int> dice(1,6);
uniform_real_distribution<double> score(0.0,10.0);

const int N = 15; 
ll fact[N+1];

void set_fact() {
    fact[0] = 1;
    rep(i,N) fact[i+1] = ((ll)i+1) * fact[i];
}

const int D = 10;
int cnt[D];
int n;
string in;
ll inm, ins;

ll calc_parm(){
    int sum = 0;
    rep(i, D) sum += cnt[i];

    ll ret = fact[sum];
    rep(i, D){
        ret /= fact[cnt[i]];
    }

    return ret;
}

ll calc(){
    vi cnt_org = cnt;

    ll ret = 0LL;
    rep(i, n){
        int num = in[i] - '0';

        rep(j, num){
            cnt[j]--;
            ret += calc_parm();
            cnt[j]++;
        }

        if(cnt[num] <= 0) break;
        cnt[num]--;
    }

    cnt = cnt_org;

    return ret;
}

ll dfs(int num, int depth, ll mul, ll sum){
    if(sum > ins) return 0LL;

    if(depth == n){
        if(sum <  ins) return calc_parm();
        if(sum == ins){
            if(mul <  inm) return calc_parm();
            if(mul == inm){
                return calc();
            }
        }
        return 0LL;
    }

    ll ret = 0LL;
    rep(nnum, num, 10){
        cnt[nnum]++;
        ret += dfs(nnum, depth + 1, mul * (nnum + 1), sum + nnum);
        cnt[nnum]--;
    }

    return ret;
}

int main(void){
    in; cin >> in;
    n = in.size();

    set_fact();

    ins = 0LL, inm = 1LL;
    for(auto & c : in){
        int e = c - '0';
        ins += e;
        inm *= (e + 1);
    }

    cout << dfs(0, 0, 1LL, 0LL) << endl;

    return 0;
}

Compiler output:
a.cc: In function 'll calc()':
a.cc:80:18: error: conversion from 'int [10]' to non-scalar type 'vi' {aka 'std::vector<int>'} requested
   80 |     vi cnt_org = cnt;
      |                  ^~~
a.cc:96:9: error: incompatible types in assignment of 'vi' {aka 'std::vector<int>'} to 'int [10]'
   96 |     cnt = cnt_org;
      |     ~~~~^~~~~~~~~