If we choose 3 different numbers from 2, 4, 6, 7, 8 and 9 to form 3-digit even numbers, how many of these numbers is / are smaller than 783?

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.

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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.

Explore how to solve the problem of forming three-digit even numbers using digits 2, 4, 6, 7, 8, and 9 that are smaller than 783. Learn the systematic approach and calculations involved. Suitable for students in Grades 5-7.

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