Kertoma

import java.math.BigInteger;
import java.util.Objects;
import java.util.Scanner;

public class Factorial2 {
    public static void main(String[] args) {
        Scanner input = new Scanner(System.in);
        int[] occurrences = new int[10];
        int numberCount = 0;
        int i = 0;
        String str = input.nextLine();
        for (String s : str.split(" ")) {
            int value = Integer.parseInt(s);
            numberCount += value;
            occurrences[i++] = value;
        }

        if (numberCount <= 3 && checkSmallFactorial(occurrences)) return;
        checkBigFactorial(numberCount);
    }

    /**
     * Detects 2! - 6! (both inclusive)
     */
    private static boolean checkSmallFactorial(int[] occurrences) {
        int fact = 1;
        for (int i = 2; i < 7; i++) {
            fact *= i;
            if (isSolution(String.valueOf(fact), occurrences.clone())) {
                System.out.println(i);
                return true;
            }
        }
        return false;
    }

    private static boolean isSolution(String candidate, int[] occurrences) {
        for (int num = 0; num < 10; num++) {
            int occurrencesOfNumber = 0;
            char numberChar = Character.forDigit(num, 10);
            for (char c : candidate.toCharArray()) {
                if (numberChar == c) occurrencesOfNumber++;
            }
            if (occurrences[num] != occurrencesOfNumber) return false;
        }
        return true;
    }

    /**
     * Detects 7! and greater based on length
     */
    private static void checkBigFactorial(int numberCount) {
        BigInteger num = BigInteger.valueOf(6);
        BigInteger fact = BigInteger.valueOf(720);
        while (numberCount > fact.toString().length()) {
            fact = fact.multiply(num = num.add(BigInteger.ONE));
        }

        System.out.println(num);
    }
}

Test 1

Group: 1, 2, 3

Verdict: ACCEPTED

Test 2

Group: 1, 2, 3

Verdict: ACCEPTED

Test 3

Group: 1, 2, 3

Verdict: ACCEPTED

Test 4

Group: 1, 2, 3

Verdict: ACCEPTED

Test 5

Group: 1, 2, 3

Verdict: ACCEPTED

Test 6

Group: 1, 2, 3

Verdict: ACCEPTED

Test 7

Group: 2, 3

Verdict: ACCEPTED

Test 8

Group: 2, 3

Verdict: ACCEPTED

Test 9

Group: 2, 3

Verdict: ACCEPTED

Test 10

Group: 2, 3

Verdict: ACCEPTED

Test 11

Group: 3

Verdict: TIME LIMIT EXCEEDED

Test 12

Group: 3

Verdict: TIME LIMIT EXCEEDED

Test 13

Group: 3

Verdict: TIME LIMIT EXCEEDED

Test 14

Group: 3

Verdict: TIME LIMIT EXCEEDED

Test 15

Group: 3

Verdict: TIME LIMIT EXCEEDED

Test 16

Group: 3

Verdict: TIME LIMIT EXCEEDED