Forming 6-Digit Numbers from the Digits 1 to 8 Without Repetition

When forming 6-digit numbers using the digits from 1 to 8 without repetition, a systematic approach can be utilized to determine the total number of unique combinations. This article will delve into the methodology and provide a comprehensive understanding of the process.

Understanding the Problem

The problem at hand involves forming 6-digit numbers using the digits 1, 2, 3, 4, 5, 6, 7, and 8, with the restriction that each digit can only be used once. This means that the number of digits available decreases with each choice, leading to a factorial-based solution.

Step-by-Step Solution

Let’s break down the process step-by-step to ensure clarity and accuracy.

1. Choosing the Most Significant Digit (MSD)

For the most significant digit (the first digit in a 6-digit number), there are 8 possible choices (from 1 to 8). Once a digit is chosen, it cannot be reused.

2. Choosing Subsequent Digits

For the second digit, there are 7 remaining choices (since one digit was already used). Similarly, 6 choices remain for the third digit, 5 for the fourth, 4 for the fifth, and 3 for the sixth digit.

3. Calculating the Total Number of Combinations

The total number of unique 6-digit numbers can be calculated by multiplying the number of choices for each digit:

Total Number of Combinations 8 * 7 * 6 * 5 * 4 * 3

Performing the multiplication:

8 * 7 56
56 * 6 336
336 * 5 1680
1680 * 4 6720
6720 * 3 20160

Alternative Method: Factorial Formula

A more concise method to determine the number of unique 6-digit combinations is by using the permutation formula for combinations without repetition, which is given by:

N! / (N - C)!

Here, N represents the total number of digits available (8 in this case), and C represents the number of digits to be chosen (6 in this case).

4. Applying the Factorial Formula

Calculate N! (the factorial of N, which is 8!) and (N - C)! (the factorial of the result of 8 - 6, which is 2!):

8! 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 40320
2! 2 * 1 2

Now, divide the results:

40320 / 2 20160

Conclusion

In conclusion, there are a total of 20,160 unique 6-digit numbers that can be formed from the digits 1 to 8 without repetition. This systematic approach not only ensures accuracy but also illustrates a compact mathematical formula for determining the number of such permutations.