How many numbers are there between 1245
 and 5421
 inclusive which contain each of the digits 1,2,4,
 and 5
 once and once only?

Question 27Answer

a.
4,177



b.
5,421



c.
4,176



d.
42



e.
24

To solve this problem, we need to determine how many numbers between 1245 and 5421 (inclusive) contain each of the digits 1, 2, 4, and 5 exactly once.

### Step-by-step Solution:

1. **Understanding the Problem:**
   - We want to form a number using the digits 1, 2, 4, and 5.
   - Each digit must appear exactly once in each number.
   - The number must be between 1245 and 5421 (inclusive).

2. **Counting All Possible Combinations of Digits 1, 2, 4, and 5:**
   - There are 4 digits, and we need to form a 4-digit number using these digits exactly once.
   - The number of permutations of 4 distinct digits is \(4! = 24\).

3. **Finding the Permutations Within the Range [1245, 5421]:**
   - We need to check which of these 24 permutations fall between 1245 and 5421.

4. **Listing the Permutations of the Digits 1, 2, 4, and 5:**
   - The permutations of 1, 2, 4, 5 are:
     1. 1245
     2. 1254
     3. 1425
     4. 1452
     5. 1524
     6. 1542
     7. 2145
     8. 2154
     9. 2415
     10. 2451
     11. 2514
     12. 2541
     13. 4125
     14. 4152
     15. 4215
     16. 4251
     17. 4512
     18. 4521
     19. 5124
     20. 5142
     21. 5214
     22. 5241
     23. 5412
     24. 5421

5. **Counting Permutations Between 1245 and 5421 (Inclusive):**
   - By looking at the list above, we see all 24 permutations are within the range from 1245 to 5421.

### Conclusion:
- **There are 24 numbers that meet the criteria.**

Therefore, the correct answer is:

**e. 24**

Would you like a detailed explanation on how to check each permutation manually, or have any other questions? Here are some follow-up questions to consider:

1. What is the formula for calculating permutations for different numbers of digits?
2. How do you find combinations of numbers that fit a specific range?
3. What are the differences between permutations and combinations?
4. How would the answer change if the digits were not unique?
5. How would you solve this problem programmatically using a computer?

**Tip:** When solving permutation problems, listing all possible combinations helps ensure no options are missed and makes range-checking easier.

How many numbers are there between 1245 and 5421 inclusive which contain each of the digits 1, 2, 4, and 5 once and once only?

Explore how to determine the number of 4-digit numbers between 1245 and 5421 that include each digit 1, 2, 4, and 5 exactly once. This problem involves key concepts in permutations and combinatorics, suitable for students in Grades 9-12.

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