11. There is a kind of four-digit number, with each digit being one of 1, 2, 3, or 4.
If the thousands digit is not 1, the hundreds digit is not 2, the tens digit is not 3, and the ones digit is not 4, how many such four-digit numbers are there? (Write all such four-digit numbers.)

12. There is a five-digit number. Each digit is either 1, 2, 3, or 5, and adjacent digits are different. How many such five-digit numbers are there in total?

There are 81 valid four-digit numbers with given restrictions. Additionally, there are 324 unique five-digit numbers with adjacent digits being different.
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Answered by Anonymous | 2025-07-04

Let’s address each question one by one:
11. Four-digit number with digits 1, 2, 3, or 4:
Given the conditions:

The thousands digit is not 1.
The hundreds digit is not 2.
The tens digit is not 3.
The ones digit is not 4.

Step-by-step analysis:

Thousands Digit : Since it cannot be 1, the thousands digit can be 2, 3, or 4.

Hundreds Digit : Since it cannot be 2, the hundreds digit can be 1, 3, or 4.

Tens Digit : Since it cannot be 3, the tens digit can be 1, 2, or 4.

Ones Digit : Since it cannot be 4, the ones digit can be 1, 2, or 3.


To find the total number of such combinations, multiply the number of choices for each digit position:
3 (choices for thousands digit) × 3 (choices for hundreds digit) × 3 (choices for tens digit) × 3 (choices for ones digit) = 81
So, there are 81 such four-digit numbers.
12. Five-digit number with digits 1, 2, 3, or 5:
Given the condition that adjacent digits are different:
Step-by-step analysis:

First Digit Choice : For the first digit, there are 4 possible choices: 1, 2, 3, or 5.

Subsequent Digits Choice : For each subsequent digit, there must be a different digit compared to the previous one. Thus, there are 3 choices for each digit after the first.

Calculation : To find the total number of five-digit numbers:

4 choices for the first digit
3 choices each for the next four digits



Therefore:
4 × 3 × 3 × 3 × 3 = 324
So, there are 324 such five-digit numbers.
In conclusion, utilizing basic counting principles and constraints given by the problem highlights the logical approach necessary to determine the number of valid combinations for both scenarios. These exercises are excellent for understanding combinatorics and logical reasoning.

Answered by LiamAlexanderSmith | 2025-07-06