Counting Distinct 4-Digit Numbers Greater Than 5000 from Given Digits

Counting the number of distinct 4-digit numbers greater than 5000 that can be formed from a specific set of digits involves a detailed analysis. This process highlights the importance of permutations, leading digit constraints, and combinatorial mathematics in educational and practical scenarios. In this article, we will explore the methodology to solve such problems and provide a detailed breakdown of the steps involved.

Given the digits 3 3 5 5 5 8 8 8 8, we need to determine how many distinct 4-digit numbers can be formed such that the number is greater than 5000. Let's break this down:

Case 1: Leading Digit is 5

If the leading digit is 5, we need to choose 3 more digits from the remaining digits: 3 3 5 8 8 8 8. We will consider two subcases based on the presence of the digit 5.

Subcase 1.1: Choosing Three Digits with At Least One Digit as 5

3 3 5: Arrangements (frac{3!}{2!} 3) 3 5 8: Arrangements (3! 6) 5 8 8: Arrangements (frac{3!}{2!} 3) 5 5 8: Arrangements (frac{3!}{2!} 3) (Note: Arrangements for each combination include permutations of the digits)

Total for this subcase: (3 6 3 3 15)

Subcase 1.2: Choosing Three Digits with No Digit as 5

3 8 8: Arrangements (frac{3!}{2!} 3)

Total for this case: (15 3 18)

Case 2: Leading Digit is 8

For the leading digit being 8, we need to choose 3 more digits from the remaining digits: 3 3 5 5 5 8 8 8.

3 3 5: Arrangements (frac{3!}{2!} 3) 3 5 5: Arrangements (frac{3!}{2!} 3) 3 8 8: Arrangements (frac{3!}{2!} 3) 5 5 8: Arrangements (frac{3!}{2!} 3) 5 8 8: Arrangements (frac{3!}{2!} 3) (Note: Arrangements for each combination include permutations of the digits)

Total for this case: (3 3 3 3 3 15)

Total Distinct 4-Digit Numbers Greater Than 5000

Now, summing the totals from both cases:

Total for leading digit 5: 18 Total for leading digit 8: 15

Total 18 15 33

Thus, the total number of distinct 4-digit numbers that can be formed using the given digits and greater than 5000 is 33.

Alternative Approach

Another approach involves considering all valid 4-digit combinations that satisfy the conditions that the digit '3' cannot appear more than twice, '5' cannot appear more than three times, and '8' cannot appear more than four times. The following table outlines these valid combinations and their permutations:

Combination Permutations 5533 4!/2!2!/2 6/2 3 5338 4!/2!2 6 5388 4!/2!3/4 9 5553 4!/3! 3 5558 4!/3! 4 5888 4!/3! 4 8553 4!/2!3/4 9 8833 [4!/2!2!]/2 3 8883 [4!/3!]3/4 3 8888 4!/4! 1

Total 3 6 9 3 4 4 9 3 3 1 51

The final count from this approach is 51, but we must exclude the invalid combinations (5333, 5555, and 8333), reducing the total to 51 - 3 48.

Thus, the final count of distinct 4-digit numbers greater than 5000 is 48.

This problem emphasizes the application of permutations, leading digit constraints, and combinatorial methods in mathematical problem-solving, making it a valuable learning tool for students and professionals alike in fields such as mathematics, statistics, and computer science.