Constructing Four-Digit Even Numbers with Distinct Digits

This article explores the problem of constructing four-digit even numbers using the digits 0, 1, 2, 3, 4, 5, and 6 with no repeated digits. We will illustrate the solution using combinatorial methods and validate our results with a sample code snippet written in J programming language.

Introduction

When constructing numbers, constraints such as ensuring all digits are distinct and that the number is even add complexity to the problem. This article delves into the process of constructing four-digit even numbers under these conditions, providing a step-by-step explanation and verification.

Combinatorial Approach

To form a four-digit even number using the digits 0 to 6 with no repeated digits, the last digit must be one of 0, 2, or 4. We will break the problem into two cases based on the last digit:

Case 1: The Number Ends in 0

When the last digit is 0:

There is 1 way to choose the last digit (0). 5 ways to choose the first digit (any of the remaining digits except 0). 4 ways to choose the second digit (from the remaining digits). 3 ways to choose the third digit (from the remaining digits).

Therefore, the total number of such four-digit numbers is:

1 * 5 * 4 * 3 60

Case 2: The Number Ends in 2 or 4

When the last digit is 2 or 4:

2 ways to choose the last digit (either 2 or 4). 4 ways to choose the first digit (any of the remaining digits except 0, and the digit that was chosen for the last digit). 4 ways to choose the second digit (from the remaining digits). 3 ways to choose the third digit (from the remaining digits).

Therefore, the total number of such four-digit numbers is:

2 * 4 * 4 * 3 96

Adding both cases, the total number of four-digit even numbers with distinct digits is:

60 96 156

Verification Using J Programming Language

The J programming language can be used to brute force the solution to this problem. The following code snippet lists all 156 such four-digit numbers:

4432 48. The first number in that product is 4 not 5 because the leading digit of a number cannot be 0. Brute force solution using the J programming language: n./:~ev~1000 10.4 perm 6 156 The answer is 156 positive even integers. List them: n 1024 1032 1034 1042 1052 1054 1204 1230 1234 1240 1250 1254 1302 1304 1320 1324 1340 1342 1350 1352 1354 1402 1420 1430 1432 1450 1452 1502 1504 1520 1524 1530 1532 1534 1540 1542 2014 2034 2054 2104 2130 2134 2140 2150 2154 2304 2310 2314 2340 2350 2354 2410 2430 2450 2504 2510 2514 2530 2534 2540 3012 3014 3024 3042 3052 3054 3102 3104 3120 3124 3140 3142 3150 3152 3154 3204 3210 3214 3240 3250 3254 3402 3410 3412 3420 3450 3452 3502 3504 3510 3512 3514 3520 3524 3540 3542 4012 4032 4052 4102 4120 4130 4132 4150 4152 4210 4230 4250 4302 4310 4312 4320 4350 4352 4502 4510 4512 4520 4530 4532 5012 5014 5024 5032 5034 5042 5102 5104 5120 5124 5130 5132 5134 5140 5142 5204 5210 5214 5230 5234 5240 5302 5304 5310 5312 5314 5320 5324 5340 5342 5402 5410 5412 5420 5430 5432

Conclusion

The solution to the problem of constructing four-digit even numbers with distinct digits requires careful consideration of the constraints on the digits and their placement. Using combinatorial methods and validating the results with code ensures accuracy and completeness.

Understanding these techniques can be particularly useful in solving similar problems related to permutation and combination, and the application of programming techniques like the J programming language can enhance the problem-solving process.