To form a 4-digit even number using the digits 2, 3, 4, 6, 7, 8, and 9 without repeating any digit, we first identify the available even digits in our set. The even digits from the list are 2, 4, 6, and 8.
Step 1: Choosing the last digit
Since we need the number to be even, the last digit of the 4-digit number must be one of the even digits. We can choose from the following 4 even digits: 2, 4, 6, 8.
We have 4 options for the last digit.
Step 2: Choosing the first three digits
After selecting the last digit, we will have 6 remaining digits available to choose from.
The options for the first three digits can be selected as follows:
- Choose the last digit: Assume we selected one of the even digits. There are 4 options: (2, 4, 6, or 8).
- Choose the first digit: We can choose any of the remaining 6 digits except the one we selected as the last digit.
- Choose the second digit: We then choose from the remaining 5 digits.
- Choose the third digit: Finally, we can choose from the remaining 4 digits.
Step 3: Calculating the total
For each choice of the last digit, the number of ways to choose the first three digits can be calculated by multiplying the number of available choices:
- Choose the last digit: 4 choices (2, 4, 6, or 8).
- Choose the first digit (still 6 choices): 6 options.
- Choose the second digit (now 5 choices left): 5 options.
- Choose the third digit (4 choices remaining): 4 options.
We perform the calculation: \[ \text{Total combinations} = \text{(Choices for last digit)} \times \text{(Choices for 1st digit)} \times \text{(Choices for 2nd digit)} \times \text{(Choices for 3rd digit)} \] \[ = 4 \times 6 \times 5 \times 4 \]
Calculating the above: \[ 4 \times 6 = 24 \] \[ 24 \times 5 = 120 \] \[ 120 \times 4 = 480 \]
Thus, the total number of 4-digit even numbers that can be formed using the digits 2, 3, 4, 6, 7, 8, and 9 without repetition is \(\boxed{480}\).
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