Question 5: An unknown code consists of 8 characters. Each character can be a letter or a digit. There are therefore 36 usable characters for each position.
A code-breaking software tests approximately one hundred million codes per second. In how much time at most can the software discover the code?

\begin{tabular}{l} a. approximately 0.3

seconds

& b. approximately 8 hours & c. approximately 3 hours &

d. approximately 470

hours

\hline \end{tabular}

A bank asked its customers to create a personal six-digit password, formed only by digits from 0 to 9, for access to their checking account via the internet.
However, an expert in electronic security systems recommended to the bank's management to re-register its users, requesting, for each one of them, the creation of a new six-digit password, now allowing the use of the 26 letters of the alphabet, in addition to digits from 0 to 9. In this new system, each uppercase letter was considered distinct from its lowercase version. Furthermore, the use of other types of characters was prohibited.
One way to evaluate a change in the password system is to verify the improvement coefficient, which is the ratio of the new number of password possibilities to the old one.
The improvement coefficient of the recommended change is
(A) $\frac{62^{6}}{10^{6}}$ (B) $\frac{62!}{10!}$ (C) $\frac{62! \cdot 4!}{10! \cdot 56!}$ (D) $62! - 10!$ (E) $62^{6} - 10^{6}$

A company will build its page on the internet and expects to attract an audience of approximately one million customers. To access this page, a password with a format to be defined by the company will be required. There are five format options offered by the programmer, described in the table, where ``L'' and ``D'' represent, respectively, uppercase letter and digit.

Option	Format
I	LDDDDD
II	DDDDDD
III	LLDDDD
IV	DDDDD
V	LLLDD

The letters of the alphabet, among the 26 possible ones, as well as the digits, among the 10 possible ones, can be repeated in any of the options.
The company wants to choose a format option whose number of possible distinct passwords is greater than the expected number of customers, but such that this number is not greater than twice the expected number of customers.
The option that best suits the company's conditions is
(A) I.
(B) II.
(C) III.
(D) IV.
(E) V.

Let S be the set of all 5-digit numbers that contain the digits $1,3,5,7$ and 9 exactly once (in usual base 10 representation). Show that the sum of all elements of S is divisible by 11111. Find this sum.

The format for car license plates in a small country is two digits followed by three vowels, e.g. 04 IOU. A license plate is called ``confusing'' if the digit 0 (zero) and the vowel O are both present on it. For example $04\,IOU$ is confusing but $20\,AEI$ is not. (i) How many distinct number plates are possible in all? (ii) How many of these are not confusing?

Among the numbers $1,2,3,4,5$, if we select four numbers with repetition allowed and arrange them in a row to form a four-digit natural number that is a multiple of 5, how many cases are there? [3 points]
(1) 115
(2) 120
(3) 125
(4) 130
(5) 135

From the numbers $1,2,3,4,5,6$, select five numbers with repetition allowed and arrange them in a line to form a five-digit natural number, satisfying the following conditions. How many such five-digit natural numbers can be formed? [4 points] (가) Each odd number is either not selected or selected exactly once. (나) Each even number is either not selected or selected exactly twice.

Among four-digit natural numbers that can be formed by selecting 4 numbers from the digits 1, 2, 3, 4, 5 with repetition allowed and arranging them in a line, how many are odd numbers greater than or equal to 4000? [3 points]
(1) 125
(2) 150
(3) 175
(4) 200
(5) 225

6. Using the digits $0, 1, 2, 3, 4, 5$ to form five-digit numbers with no repeated digits, the number of even numbers greater than $40000$ is
(A) $144$
(B) $120$
(C) $96$
(D) $72$

The sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4$, and 5, each digit appearing at most once, is
(A) 399900
(B) 399960
(C) 390000
(D) 360000

Find the number of six-digit numbers using digits from $\{2, 3, 9\}$ (repetition allowed) that are divisible by 6.
(A) 80 (B) 82 (C) 81 (D) 83

Find the sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4,5$, each digit appearing at most once.

Find the sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4,5$, each digit appearing at most once.

The sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4$, and 5, each digit appearing at most once, is
(A) 399900
(B) 399960
(C) 390000
(D) 360000

The sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4$, and 5, each digit appearing at most once, is
(A) 399900
(B) 399960
(C) 390000
(D) 360000

How many numbers formed by rearranging the digits of 234578 are divisible by 55?
(A) 0
(B) 12
(C) 36
(D) 72

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is
(A) 55
(B) 66
(C) 77
(D) 88

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $0,2,3,4,6,7$ is $\_\_\_\_$.

Let $S$ be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in $S$, but 0210222 is NOT in $S$.
Then the number of elements $x$ in $S$ such that at least one of the digits 0 and 1 appears exactly twice in $x$, is equal to $\_\_\_\_$ .

The number of 3-digit numbers, with distinct digits, that can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ and divisible by 3 is
(1) 80
(2) 120
(3) 40
(4) 108

5-digit numbers are to be formed using $2,3,5,7,9$ without repeating the digits. If $p$ be the number of such numbers that exceed 20000 and $q$ be the number of those that lie between 30000 and 90000, then $p : q$ is:
(1) $6 : 5$
(2) $3 : 2$
(3) $4 : 3$
(4) $5 : 3$

The sum of the digits in the unit's place of all the 4-digit numbers formed by using the numbers $3,4,5$ and $6$, without repetition is:
(1) 18
(2) 36
(3) 108
(4) 432

The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is:
(1) 216
(2) 192
(3) 120
(4) 72

The number of integers greater than 6000 that can be formed, using the digits $3,5,6,7$ and 8 , without repetition is
(1) 72
(2) 216
(3) 192
(4) 120

The number of numbers between 2,000 and 5,000 that can be formed with the digits $0,1,2,3,4$ (repetition of digits is not allowed) and are multiple of 3 is
(1) 36
(2) 30
(3) 24
(4) 48