Question

How many even numbers of at least four digits can be formed using the digits \(0,1,2,3,\) and 5 without repetitions?

Short answer

42

Step-by-step solution

- Identify digit constraints

The number must be at least four digits long and formed from the digits 0, 1, 2, 3, and 5 without repetition.

- Determine possible last digits

Since the number must be even, the last digit must be 0 or 2.

- Calculate when last digit is 0

If the last digit is 0, the other three digits must be selected from 1, 2, 3, and 5. There are 4 choices for the first digit (cannot be 0), 3 choices for the second digit, and 2 choices for the third digit: The number of ways = 4 * 3 * 2 = 24.

- Calculate when last digit is 2

If the last digit is 2, the other three digits must be selected from 0, 1, 3, and 5. There are 3 choices for the first digit (cannot be 0), 3 choices for the second digit, and 2 choices for the third digit: The number of ways = 3 * 3 * 2 = 18.

- Add the results

Sum the number of ways to form the number with both possible scenarios for the last digit.Total ways = 24 (when last digit is 0) + 18 (when last digit is 2) = 42.

Key concepts

Digit Constraints

When forming numbers under specific rules, it's essential to grasp the digit constraints. In this exercise, we're working with the digits 0, 1, 2, 3, and 5. To form valid numbers:

• They must be at least four digits long.
• No digit can be repeated.
• Since the numbers must be even, the last digit must be one of the even digits: 0 or 2.

These constraints ensure that the numbers meet the problem requirements. By understanding and applying these rules, we can accurately determine valid number combinations.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arrangement, and combination of objects. In this problem, we need to count how many valid combinations of digits can form four-digit even numbers.
First, we separate possibilities by fixing the last digit (since the number is even, it must be 0 or 2).
Using combinatorics, we calculate the number of valid configurations for each scenario to find the total number of combinations. This structured approach simplifies complex counting problems by breaking them down into manageable steps.

Permutations

Permutations involve arranging all elements of a set in specific orders. Here, we're arranging digits to form numbers. Permutations become restricted by digit constraints and the need for the digit order to follow specific rules.
Let's take when the last digit is 0:
We have to choose 3 more digits from the remaining 4 digits (1, 2, 3, 5) without allowing 0 as the first digit. Thus, we calculate:

• 4 choices for the first digit (1, 2, 3, 5)
• 3 choices for the second digit
• 2 choices for the third digit

This calculation gives 4 * 3 * 2 = 24 permutations.
Similarly, we calculate permutations when the last digit is 2, leading to another set of permutations.

Step-by-Step Problem Solving

Breaking down problems into individual steps ensures we don't miss any details. Let's review the step-by-step process:

1. Identifying digit constraints establishes the rules for valid numbers.
2. Fixing the last digit as either 0 or 2 simplifies permutations calculations.
3. For each scenario (last digit 0 or 2), calculate possible permutations of remaining digits.
4. Finally, add the results for total valid combinations.

This step-by-step approach clarifies complex problems, making it easier to find accurate solutions. For instance, we calculated 24 ways for the last digit being 0 and 18 ways for the last digit being 2, giving a total of 42 ways to form valid numbers.