Forming 4-Digit Numbers Greater Than 5000 from 1, 2, 3, 4, 5, 6, and 7
In this article, we will explore how to form 4-digit numbers greater than 5000 using the digits 1, 2, 3, 4, 5, 6, and 7, with each digit being unique. This problem involves combinatorial mathematics and permutations. We will provide a step-by-step solution to find the total number of such 4-digit numbers.
Understanding the Problem
Our goal is to determine how many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, and 7 without repetition, and ensuring that each number is greater than 5000.
Step-by-Step Solution
Step 1: Determine the First Digit
For a 4-digit number to be greater than 5000, the first digit must be either 5, 6, or 7. Let's analyze each case separately.
Case 1: First Digit is 5
After choosing 5 as the first digit, we need to select 3 more digits from the set {1, 2, 3, 4, 6, 7}. The number of ways to choose 3 digits out of 6 is calculated using combinations:
(binom{6}{3} 20)
Once the 3 digits are chosen, they can be permuted in (3! 6) ways. Therefore, the total number of valid 4-digit numbers for this case is:
20 * 6 120
Case 2: First Digit is 6
Similarly, if the first digit is 6, we need to choose 3 more digits from {1, 2, 3, 4, 5, 7}. The calculation is identical to Case 1:
(binom{6}{3} 20)
Again, the 3 digits can be permuted in (3! 6) ways. The total for this case is:
20 * 6 120
Case 3: First Digit is 7
If the first digit is 7, we need to choose 3 more digits from {1, 2, 3, 4, 5, 6}. The calculation is the same as in the previous cases:
(binom{6}{3} 20)
And the 3 digits can be permuted in (3! 6) ways, resulting in:
20 * 6 120
Step 3: Summing Up the Totals
Now, we sum the totals from all three cases:
120 (from Case 1) 120 (from Case 2) 120 (from Case 3) 360
Therefore, the total number of 4-digit numbers with different digits greater than 5000 that can be formed from the digits 1, 2, 3, 4, 5, 6, and 7 is 360.
Brute Force Solution Using J Programming Language
For those interested in a computational approach, we can use the J programming language to verify our solution. The code provided below demonstrates the brute force method to find the number of 4-digit numbers greater than 5000 with or without allowing digit duplication.
Case 1: Allowed digit duplication
n.~500010.odo 45{1 2 3 5 7250
The answer is 250 four-digit integers greater than 5000 if digit duplication is allowed.
Case 2: Disallowed digit duplication
n.~500010.4 perm 5{1 2 3 5 748
The answer is 48 four-digit integers greater than 5000 if digit duplication is NOT allowed.
Listing all the numbers for Case 2 when duplication is disallowed:
n5123 5132 5213 5231 5312 5321 7123 7132 7213 7231 7312 7321 5127 5172 5217 5271 5712 5721 7125 7152 7215 7251 7512 7521 5137 5173 5317 5371 5713 5731 7135 7153 7315 7351 7513 7531 5237 5273 5327 5372 5723 5732 7235 7253 7325 7352 7523 7532
Conclusion
By following this combinatorial approach, we have determined that there are 360 unique 4-digit numbers greater than 5000 that can be formed from the digits 1, 2, 3, 4, 5, 6, and 7. For a deeper understanding, consider exploring the J programming language for further computational verification.