QuestionFind all three-digit even numbers using repeating or non-repeating digits.

Studdy Solution

STEP 1

Assumptions1. A three-digit number ranges from100 to999. . An even number ends with0,,4,6, or8.
3. Repeating digits means a digit can appear more than once in a number.
4. Non-repeating digits means a digit can only appear once in a number.

STEP 2

First, let's find all possible three-digit even numbers that can be created using repeating digits.1. The hundreds place can be filled by any of the9 digits (1 to9).
2. The tens place can be filled by any of the10 digits (0 to9). . The units place (to make the number even) can be filled by any of the5 digits (0,2,4,6,8).
So, the total number of three-digit even numbers with repeating digits is given by the product of the number of choices for each place.
$Total\, repeating = Choices\, for\, hundreds\, place \\times Choices\, for\, tens\, place \\times Choices\, for\, units\, place$

STEP 3

Now, plug in the values for the number of choices for each place to calculate the total number of three-digit even numbers with repeating digits.
$Total\, repeating =9 \\times10 \\times5$

STEP 4

Calculate the total number of three-digit even numbers with repeating digits.
$Total\, repeating =9 \\times10 \\times =450$

STEP 5

Next, let's find all possible three-digit even numbers that can be created using non-repeating digits.
1. The hundreds place can be filled by any of the9 digits (1 to9).
2. The tens place can be filled by any of the9 remaining digits (0 and the8 other digits not used in the hundreds place).
3. The units place (to make the number even) can be filled by any of the4 remaining even digits (not used in the hundreds and tens places).
So, the total number of three-digit even numbers with non-repeating digits is given by the product of the number of choices for each place.
$Total\, non-repeating = Choices\, for\, hundreds\, place \\times Choices\, for\, tens\, place \\times Choices\, for\, units\, place$

STEP 6

Now, plug in the values for the number of choices for each place to calculate the total number of three-digit even numbers with non-repeating digits.
$Total\, non-repeating =9 \\times9 \\times4$

STEP 7

Calculate the total number of three-digit even numbers with non-repeating digits.
$Total\, non-repeating =9 \\times9 \\times4 =324$ So, there are450 three-digit even numbers that can be created using repeating digits, and324 three-digit even numbers that can be created using non-repeating digits.

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