Numbers Analysis: Counting 3-Digit Numbers with Restrictions

This problem asks us to find the total count of 3-digit numbers where the digit 1 is never positioned immediately to the right of the digit 2. This means any 3-digit number containing the sequence "21" (like 210, 321, etc.) is excluded from our count. To solve this, we will first find the total number of 3-digit numbers and then subtract the count of numbers that contain the forbidden sequence "21".

Total 3-Digit Numbers

A 3-digit number ranges from 100 to 999. Let's determine how many such numbers exist.

• The hundreds digit cannot be 0, so it can be any digit from 1 to 9. This gives us 9 choices.
• The tens digit can be any digit from 0 to 9. This gives us 10 choices.
• The units digit can be any digit from 0 to 9. This also gives us 10 choices.

Therefore, the total number of 3-digit numbers is calculated as:

$ \text{Total 3-digit numbers} = \text{(Choices for hundreds digit)} \times \text{(Choices for tens digit)} \times \text{(Choices for units digit)} $

$ \text{Total 3-digit numbers} = 9 \times 10 \times 10 = 900 $

Identifying Forbidden Sequences: '21'

The condition states that the digit 1 is never to the immediate right of 2. This means any 3-digit number containing the sequence "21" is forbidden. We need to find all such 3-digit numbers and subtract them from the total. The "21" sequence can appear in two possible positions within a 3-digit number:

1. The hundreds digit is 2 and the tens digit is 1 (form: 21X).
2. The tens digit is 2 and the units digit is 1 (form: X21).

Case 1: '21' in Hundreds and Tens Place (21X)

In this case, the number has the form 21X, where X represents the units digit.

• The hundreds digit is fixed as 2.
• The tens digit is fixed as 1.
• The units digit (X) can be any digit from 0 to 9. There are 10 possibilities for X.

The numbers in this category are: 210, 211, 212, 213, 214, 215, 216, 217, 218, 219.

Total numbers in Case 1 = 10.

Case 2: '21' in Tens and Units Place (X21)

In this case, the number has the form X21, where X represents the hundreds digit.

• The tens digit is fixed as 2.
• The units digit is fixed as 1.
• The hundreds digit (X) can be any digit from 1 to 9 (since it's a 3-digit number, it cannot be 0). There are 9 possibilities for X.

The numbers in this category are: 121, 221, 321, 421, 521, 621, 721, 821, 921.

Total numbers in Case 2 = 9.

Avoiding Overlaps in '21' Forbidden Numbers

We need to check if there is any overlap between Case 1 (numbers like 21X) and Case 2 (numbers like X21).

• Numbers from Case 1 all start with '21'. For example, 210, 211, ..., 219.
• Numbers from Case 2 all end with '21'. For example, 121, 221, ..., 921.

A number cannot simultaneously start with '21' and end with '21' in a 3-digit format (e.g., a number like 2121 would be required, which is a 4-digit number). Therefore, there is no overlap between the numbers counted in Case 1 and Case 2. These two sets of forbidden numbers are mutually exclusive.

Total number of forbidden 3-digit numbers (containing "21") = Total from Case 1 + Total from Case 2

$ \text{Total forbidden numbers} = 10 + 9 = 19 $

Final Calculation: Valid 3-Digit Numbers

To find the number of 3-digit numbers where the digit 1 is never to the immediate right of 2, we subtract the forbidden numbers from the total number of 3-digit numbers.

$ \text{Required 3-digit numbers} = \text{Total 3-digit numbers} - \text{Total forbidden numbers} $

$ \text{Required 3-digit numbers} = 900 - 19 = 881 $

Therefore, there are 881 three-digit numbers such that the digit 1 is never immediately to the right of 2.