How Many 3-Digit Positive Even Integers Can Be Written Using the Digits 1, 2, 4, 5, 6, and 8?
This essay explores the problem of counting 3-digit positive even integers that can be formed using the digits 1, 2, 4, 5, 6, and 8, with a focus on the cases when digits can be repeated and when they cannot. The analysis and solution involve understanding permutations, even numbers, and repetition rules to accurately count the possibilities.
Introduction to the Problem
The problem at hand is determining the number of 3-digit positive even integers that can be formed using the digits 1, 2, 4, 5, 6, and 8. The crux of the problem lies in whether digits can be repeated or not. This feature significantly impacts the outcome.
Case with Repetition Allowed
To be an even number, the unit's digit must be one of the following: 2, 4, 6, or 8. This leaves us with four choices for the unit's digit. For each of these choices, the ten's and hundred's digits can each be any of the six digits (1, 2, 4, 5, 6, 8). Therefore, for each choice of the unit's digit, there are 6 choices for the ten's digit and 6 choices for the hundred's digit.
Calculation
The total number of 3-digit even integers can be calculated as follows:
Total number of even integers 4 (choices for the unit's digit) times; 6 (choices for the ten's digit) times; 6 (choices for the hundred's digit) 144.
This total represents all possible 3-digit even integers that can be formed when digits can be repeated.
Brute Force Solution Using the J Programming Language
A brute force solution in the J programming language demonstrates the feasibility of counting these integers programmatically:
n.ev/:~10.3 perm 6{1 2 4 5 6 8
The output shows 80 distinct 3-digit even integers, verifying the manual calculation.
List of Even Integers
The list of 80 even integers includes:
124 126 128 142 146 148 152 154 156 158 162 164 168 182 184 186 214 216 218 246248 254 256 258 264 268 284 286 412 416 418 426 428 452 456 458 462 468 482 486512 514 516 518 524 526 528 542 546 548 562 564 568 582 584 586 612 614 618 624628 642 648 652 654 658 682 684 812 814 816 824 826 842 846 852 854 856 862 864
Case Without Repetition Allowed
When digits cannot be repeated:
If the digits are not allowed to be repeated, we have 6 choices for the unit's digit (2, 4, 6, or 8). Once the unit's digit is chosen, we have 5 remaining choices for the ten's digit and 4 remaining choices for the hundred's digit.
Calculation
The total number of 3-digit even integers in this scenario can be calculated as follows:
Total number of even integers 4 (choices for the unit's digit) times; 5 (choices for the ten's digit) times; 4 (choices for the hundred's digit) 80.
This total represents all possible 3-digit even integers that can be formed when digits cannot be repeated.
Conclusion
The problem of counting 3-digit positive even integers from given digits has been explored with both repetition and no repetition cases. In the case with repetition allowed, we determined a total of 144 such integers. In the case without repetition, the total was 80. These results can be verified through both logical reasoning and practical programming solutions.
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