Problem 34 - Algebra and Trigonometry

Question

How many three-digit numbers can be formed using the digits \(0,1,2,3,4,5,6,7,8,\) and \(9 ?\) Repeated digits are allowed.

Step-by-Step Solution

Answer

900 three-digit numbers can be formed.

Step 1 - Determine the total number of digits

Identify the available digits for forming the three-digit numbers. The digits are: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This gives us a total of 10 digits.

Step 2 - Consider the hundreds place

The hundreds place of a three-digit number cannot be 0, as that would make it a two-digit number. Thus, the possible digits for the hundreds place are 1 through 9, giving us 9 options.

Step 3 - Consider the tens place

Since repeated digits are allowed, any of the 10 digits (0-9) can occupy the tens place. Thus, there are 10 options for the tens place.

Step 4 - Consider the units place

Similarly, any of the 10 digits (0-9) can occupy the units place, giving us 10 options for the units place.

Step 5 - Calculate the total number of three-digit numbers

Multiply the number of choices for each place value: Number of choices for hundreds place = 9 Number of choices for tens place = 10 Number of choices for units place = 10 Therefore, the total number of three-digit numbers is: \( 9 \times 10 \times 10 = 900 \)

Key Concepts

Number Formation

Three-digit numbers are built by selecting digits for three specific places: the hundreds place, the tens place, and the units place. Understanding how to form numbers by choosing digits for these positions plays a crucial role in number formation.

For the given exercise, we can use the digits from 0 to 9. This gives us a total of 10 digits. Each digit can be used repeatedly.

However, forming three-digit numbers has some rules. The digit in the hundreds place cannot be zero because it would turn the number into a two-digit number. This leaves us with 9 options (1 through 9) for the hundreds place.

For the tens and units places, any of the 10 digits (0 through 9) can be used, including zero. Since repeated digits are allowed, each place value can have any of the 10 digits.

Place Value

Place value helps determine the value of the number based on the position of each digit. In a three-digit number, we have three significant positions: the hundreds place, the tens place, and the units place. Each position represents a different power of 10.

For instance, in the three-digit number 245:

• The digit 2 is in the hundreds place and represents 200.
• The digit 4 is in the tens place and represents 40.
• The digit 5 is in the units place and represents 5.

In our exercise, choosing digits for these place values means we have to consider the unique restrictions for each place. The hundreds place cannot be zero due to its significant contribution to the overall value of the number. This rule limits our options for that place to the digits 1 through 9.

On the other hand, the tens and units places do not have similar restrictions, allowing any of the 10 digits from 0 to 9.

Repeated Digits

In this exercise, we are allowed to use repeated digits when forming three-digit numbers. This means that any digit from 0 to 9 can appear more than once in the number.

The allowance of repeated digits expands the total number of possible combinations. For example, both the tens and units places can use any of the 10 digits.

Let's consider why the exercise allows for repeated digits. If there were restrictions against using the same digit more than once, the count of available options would be fewer.

With repeated digits, our calculation becomes:

• 9 options for the hundreds place (since the digit cannot be zero).
• 10 options for the tens place.
• 10 options for the units place.

Thus, the total number of three-digit numbers is calculated as \[ 9 \times 10 \times 10 = 900 \] This shows how allowing repeated digits significantly increases the number of unique three-digit numbers that can be formed.