Problem 25 - Algebra and Trigonometry

Question

19–32 These problems involve permutations. Three-Digit Numbers How many different three-digit whole numbers can be formed using the digits 1, 3, 5, and 7 if no repetition of digits is allowed?

Step-by-Step Solution

Answer

There are 24 different three-digit numbers possible.

Step 1: Identify Available Digits

We have four digits available: 1, 3, 5, and 7. We need to create three-digit numbers from these digits.

Step 2: Determine Number Placement

Since each digit must be unique within a number, we have to arrange them without repetition. This means for each place we choose a digit, we reduce the choice count for the next place.

Step 3: Calculate Choices for the Hundreds Place

For the hundreds place, we can choose any of the four digits. Thus, we have 4 options.

Step 4: Calculate Choices for the Tens Place

After choosing a digit for the hundreds place, we are left with 3 remaining digits. Therefore, we have 3 choices for the tens place.

Step 5: Calculate Choices for the Units Place

After picking digits for the hundreds and tens places, only 2 digits remain. So, we have 2 choices for the units place.

Step 6: Calculate the Total Number of Permutations

Multiply the number of choices for each digit position to find the total permutations: \(4 \times 3 \times 2 = 24\).

Key Concepts

Three-Digit Numbers

Three-digit numbers are any numbers that range from 100 to 999. Since there are no repeating digits in this scenario, the exercise specifically limits our options to build these numbers using a unique set of digits. A three-digit number is structured with three positions: hundreds, tens, and units.

• The hundreds place is the first digit, and it can't be zero in this context, ensuring the number stays within three digits.
• The tens place is the second digit, contributing to shifting the range as this digit increases or decreases.
• The units place is the final digit, often changing the most as you enumerate combinations.

Grasping this concept is crucial for understanding how numbers are formed and how digit placement impacts the overall value of a number.

Digit Arrangement

Digit arrangement in the context of permutations involves systematically choosing positions for digits while adhering to particular rules. Think of it like putting digits into slots, where each slot only has one digit option at a time.

• Start by choosing a digit for the first place, among all available options.
• Then select another, ensuring the new digit isn't already used, for the next place.
• Continue this method until all places are filled.

Why is this important? Because arrangement changes the number completely. For instance, the digits 1, 3, and 5 can be arranged as 135, 153, 315, etc., each represents a different number. This sequencing process ensures you systematically cover all possibilities when arranging digits.

Non-Repetition

Non-repetition is a rule that states each digit can only appear once in a number. Imagine having a deck of cards and selecting one card at a time without putting it back. That’s similar to how digit non-repetition works.

• First, choose a digit from the pool for one of the places.
• Once chosen, that digit cannot be used again for the remainder of the number's sequence.
• This requires re-calculating the available options for every subsequent choice.

This rule of non-repetition simplifies calculations and results in a limited set of unique three-digit numbers from the available digits. For example, starting with 4 digits to fill the hundreds place, you systematically reduce available options for each subsequent position, yielding 24 unique numbers.