To solve the problem, we'll break it down into steps:

Step 1: Identify the possible 3-digit numbers

We need to form 3-digit even numbers using the digits 2, 4, 6, 7, 8, and 9.

• Hundreds digit: The first digit of the number must be from the available digits, excluding 2 and 4 because we want the number to be less than 783. So, the possible digits for the hundreds place are 2, 4, 6, and 7.
• Units digit: The last digit (units digit) must be even, since the number is even. The possible even digits from the given set are 2, 4, 6, and 8.
• Tens digit: The tens digit can be any of the remaining digits from the set, but it must be different from both the hundreds and units digits.

Step 2: Calculate cases based on the hundreds digit

We'll consider each case where the hundreds digit is less than 7 to ensure the number is smaller than 783.

Case 1: Hundreds digit is 2

• Units digit options: 4, 6, 8 (since it must be even and different from 2)

• Tens digit options: Choose from the remaining 4 digits.

  For each valid combination:

  • Units digit = 4: Tens digit options = 4 digits (6, 7, 8, 9).
  • Units digit = 6: Tens digit options = 4 digits (4, 7, 8, 9).
  • Units digit = 8: Tens digit options = 4 digits (4, 6, 7, 9).

  Total = $3 \times 4 = 12$ numbers.

Case 2: Hundreds digit is 4

• Units digit options: 2, 6, 8 (since it must be even and different from 4)

• Tens digit options: Choose from the remaining 4 digits.

  For each valid combination:

  • Units digit = 2: Tens digit options = 4 digits (6, 7, 8, 9).
  • Units digit = 6: Tens digit options = 4 digits (2, 7, 8, 9).
  • Units digit = 8: Tens digit options = 4 digits (2, 6, 7, 9).

  Total = $3 \times 4 = 12$ numbers.

Case 3: Hundreds digit is 6

• Units digit options: 2, 4, 8 (since it must be even and different from 6)

• Tens digit options: Choose from the remaining 4 digits.

  For each valid combination:

  • Units digit = 2: Tens digit options = 4 digits (4, 7, 8, 9).
  • Units digit = 4: Tens digit options = 4 digits (2, 7, 8, 9).
  • Units digit = 8: Tens digit options = 4 digits (2, 4, 7, 9).

  Total = $3 \times 4 = 12$ numbers.

Step 3: Check for the specific case where the hundreds digit is 7

Here, the number is constrained to be smaller than 783, so we need to be cautious:

• Units digit = 2: The number would be 72x (only 720 is valid; tens digit must be one of the remaining four).
• Units digit = 4: The number would be 74x (only 742 is valid).
• Units digit = 6: The number would be 76x (only 762 is valid).
• Units digit = 8: The number would be 78x, which is invalid because 78x is not smaller than 783.

Total valid combinations = 1 (720) + 1 (742) + 1 (762) = 3 numbers.

Step 4: Add all valid numbers together

Total numbers = $12 + 12 + 12 + 3 = 39$.

Final Answer:

There are 39 three-digit even numbers that can be formed with the digits 2, 4, 6, 7, 8, and 9, and are smaller than 783.

---

Would you like more details on any of the steps? Here are some related questions you might find interesting:

1. How many three-digit even numbers can be formed using the digits 1, 3, 5, 6, 7, and 9?
2. What is the probability of randomly selecting a number smaller than 500 from the set of three-digit numbers formed with the digits 1, 2, 3, 4, 5, and 6?
3. How many three-digit odd numbers can be formed using the digits 2, 4, 6, 7, 8, and 9?
4. What is the sum of all three-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5?
5. If the digits 2, 4, 6, 7, 8, and 9 are used to form 4-digit numbers, how many of these are greater than 5000?

Tip: When solving problems involving digits and place values, systematically consider each digit's constraints based on the problem requirements.