D_Even_String.cpp

#include <bits/stdc++.h>
using namespace std;
// First you make it work, then you can always make it beautiful
#define int long long
const int mod = 998244353;
const int N = 5e5 + 5;
int fact[N];
int ifact[N];

int modpowr(int a, int b) {
    int res = 1;
    while(b) {
        if(b & 1) res = (res * a) % mod;
        a = (a * a) % mod;
        b >>= 1;
    }
    return res;
}

void compute() {
    fact[0] = 1;
    for(int i=1;i<N;i++) fact[i] = (i * fact[i-1]) % mod;

    ifact[N-1] = modpowr(fact[N-1], mod - 2);
    for(int i=N-2;i>=0;i--) {
        ifact[i] = ((i+1) * ifact[i+1]) % mod;
    }

}

void solve() {
    vector<int> freq(26);
    int sum = 0;
    for(int i=0;i<26;i++) {
        cin >> freq[i];
        sum += freq[i];
    }

    int even = sum/2;
    int odd = sum - even;

    vector<int> dp(even + 1), dp2(even + 1);
    dp2[0] = 1;

    for(int i=25;i>=0;i--) {
        if(freq[i] == 0) continue;
        for(int j=0;j<=even;j++) {
            dp[j] = dp2[j];
            if(j - freq[i] >= 0) {
                dp[j] = (dp[j] + dp2[j-freq[i]]) % mod;
            }
        }
        dp2 = dp;
    }

    int ways = dp[even];

    int ans = (fact[even] * fact[odd]) % mod;
    (ans *= ways) %= mod;

    for(int i=0;i<26;i++) {
        (ans *= ifact[freq[i]]) %= mod;
    }

    cout << ans << "\n";

}


int32_t main(){
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);
    cout.tie(nullptr);

    compute();

    int _;
    cin >> _;
    while (_-->0) {
        solve();
    }

    return 0;
}