How Many Three-Digit Numbers Can Be Formed Using Digits 2, 4, 6, and 8?

When given a set of digits to form a specific number of digits, such as three-digit numbers, the methodology varies depending on whether the digits can be repeated or not. In this case, we are dealing with the digits 2, 4, 6, and 8, and each digit must be used only once. Here, we explore the solution step-by-step, delve into the mathematical reasoning, and provide a comprehensive breakdown to ensure the content is engaging for readers and optimized for SEO metrics.

Using Permutations Without Repetition

The problem asks for the total number of three-digit numbers that can be formed using the digits 2, 4, 6, and 8 without repetition. This is a classic example of finding the number of permutations with specific constraints. Let's break it down:

First, we need to choose 3 digits from the 4 available digits. The number of ways to do this is given by the combination formula binom{4}{3}:

binom{4}{3} 4

Explanation: We have 4 different choices for the first digit, 3 remaining choices for the second digit, and 2 remaining choices for the last digit, considering we do not repeat any digit. The total number of ways to arrange these chosen 3 digits is calculated by:

3! 3 times 2 times 1 6

Now, combining the two steps, we multiply the number of combinations by the number of permutations for each combination:

text{Total} binom{4}{3} times 3! 4 times 6 24

Thus, the total number of three-digit numbers that can be formed using the digits 2, 4, 6, and 8 without repetition is 24.

Calculation Method Explained

Alternatively, we can use the permutation formula directly:

4P3 frac{4!}{(4-3)!} 4! 4 times 3 times 2 times 1 24

This confirms our previous calculation. The process involves selecting 3 digits from 4 and arranging them. The 4P3 formula simplifies this process, making it easier to understand and apply.

Examples and Combinations

To further illustrate, let's enumerate the three-digit combinations:

246 248 264 268 284 286 294 296 426 428 462 468 482 486 492 496 624 628 642 648 682 684 692 694 824 826 842 846 862 864 892 894 924 926 942 946 962 964 982 984 968 969 986 989 998

Each of these combinations represents a unique three-digit number formed by using each of the digits 2, 4, 6, and 8 only once.

Use of All Digits and Repetition Allowed

It is important to note that the question could allow for repetition of digits. In such cases, we need to calculate the total number of combinations where digits can be repeated. This essentially involves selecting 3 digits from 5, with each digit potentially being the same as another.

When repetition is allowed, the total number of three-digit numbers is given by:

5^3 125

Here, each of the 3 positions in a three-digit number can be filled with any of the 5 digits, leading to 125 possible combinations.

In contrast, if no repetition is allowed, as we have already calculated, the total number of combinations is:

5 times 4 times 3 60

This means that without repetition, we have fewer unique three-digit numbers.