Question

Given the digits \(1,2,3,4,\) and \(5,\) find how many 4 -digit numbers can be formed from them: (a) If no digit may be repeated (b) If repetitions of a digit are allowed (c) If the number must be even, without any repeated digit (d) If the number must be even

Short answer

(a) 120, (b) 625, (c) 48, (d) 250

Step-by-step solution

Understand the problem

We are given the digits 1, 2, 3, 4, and 5 to form several possible 4-digit numbers under different conditions and must find the count of such numbers.

Count 4-digit numbers with unique digits (part a)

To form a 4-digit number with no repeated digits, we select and arrange 4 digits from the 5 available. The number of ways to do this is calculated by multiplying the available choices for each position: 5 choices for the first digit, 4 choices for the second, 3 for the third, and 2 for the fourth, leading to a total of \[ 5 \times 4 \times 3 \times 2 = 120 \] distinct numbers.

Count 4-digit numbers with repetition allowed (part b)

When repetition of digits is allowed, each digit in a 4-digit number can be any of the 5 available digits. Thus, the total number of such numbers is \[ 5 \times 5 \times 5 \times 5 = 5^4 = 625. \]

Count 4-digit even numbers with no repetition (part c)

For a 4-digit even number with unique digits, the last digit must be part of the even digits set {2, 4}. Calculate each case:- **Case 1:** Last digit is 2. We can pick 3 more digits from {1, 3, 4, 5}, leading to \[ 4 \times 3 \times 2 = 24 \] possibilities.- **Case 2:** Last digit is 4. We can pick 3 more digits from {1, 2, 3, 5}, leading to \[ 4 \times 3 \times 2 = 24 \] possibilities.Adding these cases yields a total of \[ 24 + 24 = 48. \]

Count 4-digit even numbers allowing repetition (part d)

For a 4-digit even number formed from digits where repetition is allowed, select the last digit as one of the even digits {2, 4} (2 choices). Each of the first three digits can be any of the 5 digits {1, 2, 3, 4, 5}, giving \[ 5 \times 5 \times 5 = 5^3 = 125 \] combinations for these digits. Therefore, the total for this case is \[ 2 \times 125 = 250. \]

Key concepts

Permutations

Permutations are fundamental in combinatorics, particularly when determining the arrangements of elements. **Permutations** refer to the different ways you can arrange a set of elements into a sequence or order. When you're forming 4-digit numbers from a set of distinct digits, like 1 through 5, and cannot repeat any digit, you're using permutations.

• The first step involves selecting which digits will be used. You begin with 5 possible choices for the first position.
• Once a digit is used, it's no longer available for the remaining positions, reducing the choices for subsequent places.
• This scenario is calculated as 5 choices for the first digit, then 4 for the second, 3 for the third, and finally 2 choices for the fourth digit.

This calculation results in a total of \[5 \times 4 \times 3 \times 2 = 120\]distinct permutations. Understanding permutations allows you to systematically count possibilities without missing a configuration.

Repetition in Combinatorics

In combinatorial problems, repetition changes the rules considerably. **Repetition in combinatorics** allows the same elements to be used multiple times in each arrangement. This contrasts with permutations without repetition, where each element is used only once.

• When repetition is permitted, each position in your sequence (number, code, etc.) can be filled with any of the available elements.
• For our specific problem, each of the four digits in a 4-digit number can be any of the 5 given digits, including duplicating digits.

This increases the total number of possible combinations, calculated as:\[5 \times 5 \times 5 \times 5 = 5^4 = 625\]Allowing repetition expands possibilities and requires an exponential approach to counting.

Even Numbers in Combinatorics

When working with numbers, forming even numbers brings into play certain conditions, as even numbers end with specific digits. For a 4-digit number to be classified as even, it must end in an even digit.

• Within the digits provided (1, 2, 3, 4, 5), the even digits are 2 and 4.
• Setting the last digit specifically to an even number simplifies counting, but introduces constraints for forming the rest of the number.
• If no repetition is allowed, once the last digit is set, the remaining digits must be selected from those that have not been used.
• Example: With the last digit fixed as 2, the other digits must be chosen from {1, 3, 4, 5} without repetition.
• When repetitions are allowed, any number for other positions can be reused, increasing the combinations significantly.

Focusing on even numbers from given sets calls for both specific choice consideration and counting strategy adjustments.

Counting Principles

Counting principles enable us to systematically and effectively count arrangements or selections. **Counting principles** form the basis of combinatorics, dictating how to multiply the choices available at different stages of the problem.

• The **fundamental counting principle** states that if there are several choices (like steps) to be made, you multiply the number of options at each step to find the total number of possibilities.
• For example, creating a 4-digit number with unique digits involves multiplying the remaining options after each selection (e.g., 5 choices, then 4, then 3, then 2).
• With repetition allowed, each position has the same number of choices (e.g., 5 choices for each position), making it a power of the total elements (e.g., 5⁴ for four positions).

Counting principles ensure that calculations, whether with repetition or without, are precise and effective, allowing us to harness mathematical structures to simplify seemingly complex problems.