Formation of Three-Digit Numbers Using Digits 1, 3, and 5: Repetition Allowed and Not Allowed

The problem of forming three-digit numbers using the digits 1, 3, and 5 can be approached in two different scenarios: repetition allowed and repetition not allowed. Let's explore each scenario in detail.

Case I: Repetition Allowed

When repetition of digits is allowed, each of the three positions in the three-digit number (hundreds, tens, and units) can be filled with any of the three given digits (1, 3, or 5).

For each position, there are 3 possible choices.

#10286; choices 3

The total number of three-digit numbers that can be formed is calculated by multiplying the number of choices for each position:

3 3

Therefore, the total number of three-digit numbers is:

27

This can be visualized as follows:

111 113 115 131 133 135 151 153 155 311 313 315 331 333 335 351 353 355 511 513 515 531 533 535 551 553 555

Case II: Repetition Not Allowed

In this scenario, each digit can only be used once in a three-digit number. Therefore, the first digit can be any of the three choices (1, 3, or 5), the second digit can be any of the two remaining choices, and the third digit will be the last remaining choice.

The number of different ways to arrange the three digits is given by the factorial of 3 (3!):

3 ! 3 * 2 * 1

Therefore, the total number of three-digit numbers that can be formed is:

6

Here is a complete list of the three-digit numbers formed without repetition:

135 153 315 351 513 531

Summary of Combinations

Tabulating all the possible three-digit combinations results in:

27

three-digit combinations when repetition is allowed, and

6

three-digit combinations when repetition is not allowed.

Conclusion and Further Exploration

Understanding how to form three-digit numbers using a set of digits with or without repetition is a fundamental concept in combinatorics. This knowledge can be applied in various fields, including mathematics, computer science, and cryptography. By exploring different scenarios, we can gain insights into the principles of permutations and combinations.