Permutations, Combinations, and Number Ranking Study of Rank-Serial Number Relationships

Problem 21: Ascending Rank Calculation
- Objective: Determine the serial number of a specific 5-digit number when a set of numbers is arranged in ascending order.
- Digit Set: The numbers are constructed from the digits $\{0, 2, 3, 4, 7, 9\}$ .
- Constraints: Repetition of digits is explicitly allowed.
- Target Number: $42923$ .
- Result: The serial number for this number is identified as $2997$ .
- Exam Reference: [ $31^{st}$ Jan. $2023$ , S-I].
Problem 22: Ranking Target 95560
- Problem Statement: Let 5-digit numbers be constructed using the digits $\{0, 3, 5, 6, 7, 9\}$ .
- Constraints: Repetition of digits is allowed.
- Arrangement: The numbers are arranged in ascending order, each associated with a unique serial number.
- Objective: Find the specific serial number corresponding to the number $95560$ .

Problem 23: Divisibility by 15
- Objective: Calculate the total number of 4-digit numbers that fulfill specific digit and divisibility criteria.
- Available Digits: $\{1, 2, 3, 5\}$ .
- Repetition: Repetition of digits is allowed.
- Divisibility Constraint: The number must be divisible by $15$ .
- Logic Overview: For a number to be divisible by $15$ , it must be divisible by both $3$ and 5 . In the given set, divisibility by $5$ mandates that the last digit must be $5$ . Divisibility by $3$ requires that the sum of the four digits equals a multiple of $3$ .
- Exam Reference: [ $30^{th}$ Jan. $2023$ , S-1].
Problem 24: Divisibility by 6
- Objective: Count the total 4-digit numbers possible given a set of digits and a divisibility rule.
- Available Digits: $\{0, 1, 2, 3, 4, 5\}$ .
- Repetition: Repetition of digits is permitted.
- Divisibility Constraint: The resulting number must be divisible by $6$ .
- Logic Overview: A number is divisible by $6$ if it is even and the sum of its digits is divisible by $3$ . Since $0$ is included in the set, the first digit cannot be $0$ . The last digit must be chosen from the set of even digits $\{0, 2, 4\}$ .

Problem 25: Vertical Descending Arrangement
- Objective: Determine the serial number of a 5-digit number when the sequence is ordered from largest to smallest.
- Digit Set: The numbers are formed using the digits $\{1, 2, 3, 5, 7\}$ .
- Repetition: Repetition of digits is allowed.
- Ascending/Descending Logic: The serial numbers are assigned in descending order.
- Example Reference: In this series, the number $77777$ is the largest possible and is assigned serial number $1$ .
- Target Evaluation: The serial number for the number $35337$ is explicitly calculated as $1436$ .
- Exam Reference: [ $29^{th}$ Jan. $2023$ , S-I].
Problem 26: Specific Rank for 43319
- Problem Statement: 5-digit numbers are formed using the digits $\{1, 3, 4, 5, 9\}$ .
- Constraints: Repetition of digits is allowed.
- Arrangement: The numbers are written in descending order with corresponding serial numbers.
- Standard Example: Using the previous logic where the highest number ( $77777$ in the transcript's placeholder example) has serial number $1$ , the sequence is established.
- Objective: Determine the exact serial number for the number $43319$ .

Documentation Series: These problems are part of the "ELITE SERIES" produced for "Sr. ICON Students".
Page Metadata: The numerical marker $44$ is present at the bottom of the source document.