Question 2

We can find the total number of possible combinations by multiplying the number of choices for each position in a code or number.

Step 1 — Understanding PIN Code Structure

Let us look at the structure of an Indian Postal Index Number (PIN) code. A PIN code is a 6-digit number. For example, 464001 for Vidisha. The first digit tells us the postal region. The second digit tells us the sub-region. The third digit tells us the sorting district. The last three digits tell us the specific post office.

Step 2 — Calculating PIN Code Combinations

Each of the 6 digits in a PIN code can be any number from 0 to 9. So, there are 10 choices for each digit position. To find the total number of theoretical combinations, we multiply the choices for each position.

Number of choices for the 1st digit = 10 Number of choices for the 2nd digit = 10 Number of choices for the 3rd digit = 10 Number of choices for the 4th digit = 10 Number of choices for the 5th digit = 10 Number of choices for the 6th digit = 10

Total theoretical combinations = $10 \times 10 \times 10 \times 10 \times 10 \times 10$

$= 10^6$

$\boxed{1,000,000 \text{ combinations}}$

Practically, only valid combinations are used based on actual geography.

Step 3 — Understanding Mobile Number Structure

Indian mobile numbers have 10 digits. They always start with the digits 6, 7, 8, or 9. The remaining 9 digits can be any number from 0 to 9.

Step 4 — Calculating Mobile Number Combinations

Let us find the total possible mobile numbers. The first digit has 4 choices (6, 7, 8, or 9). Each of the other 9 digits can be any number from 0 to 9. So, there are 10 choices for each of these 9 digits.

Number of choices for the 1st digit = 4 Number of choices for the 2nd digit = 10 Number of choices for the 3rd digit = 10 Number of choices for the 4th digit = 10 Number of choices for the 5th digit = 10 Number of choices for the 6th digit = 10 Number of choices for the 7th digit = 10 Number of choices for the 8th digit = 10 Number of choices for the 9th digit = 10 Number of choices for the 10th digit = 10

Total combinations = $4 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10$

$= 4 \times 10^9$

$\boxed{4,000,000,000 \text{ numbers}}$

Step 5 — Understanding Vehicle Registration Number Structure

Let us look at the structure of a vehicle registration number. An example is DL 01 AB 1234. The first two letters (e.g., DL) show the State or Union Territory. The next two digits (e.g., 01) show the Regional Transport Office (RTO) code. The next one or two letters (e.g., AB) show the series. We will consider two letters for the series. The last four digits (e.g., 1234) are the unique vehicle number.

Step 6 — Calculating Vehicle Registration Combinations per RTO

We want to find combinations for a single RTO. This means the State/UT code and RTO code are fixed. We need to count the combinations for the series and the unique vehicle number.

The series part has two letters. Each letter can be any of the 26 alphabets (A-Z). Number of choices for the 1st letter of the series = 26 Number of choices for the 2nd letter of the series = 26 Total letter combinations for the series = $26 \times 26 = 26^2$.

The unique vehicle number part has four digits. Each digit can be any number from 0 to 9. Number of choices for the 1st digit = 10 Number of choices for the 2nd digit = 10 Number of choices for the 3rd digit = 10 Number of choices for the 4th digit = 10 Total number combinations for the unique number = $10 \times 10 \times 10 \times 10 = 10^4$.

To find the total combinations per RTO, we multiply these two parts.

Total combinations per RTO = (Letter combinations for series) $\times$ (Number combinations for unique number)

$= 26^2 \times 10^4$

$= 676 \times 10,000$

$\boxed{6,760,000 \text{ combinations}}$

Answer

(i) PIN Codes in India: Theoretically 1,000,000 combinations. (ii) Mobile numbers in India: 4,000,000,000 possible numbers. (iii) Vehicle Registration Numbers: 6,760,000 combinations per RTO.