How Many 4-Digit Numbers Greater Than 4000 Can Be Formed Using Specific Digits?

When dealing with the formation of 4-digit numbers that are higher than 4000 using specific digits—0, 2, 3, 4, 5, and 6—under the condition that repetition is allowed, several steps must be considered. This article will guide you through the process of calculating the number of such possible combinations.

Step-by-Step Breakdown

Solving this problem involves careful consideration of the constraints and the number of choices available at each digit position. Let's break it down step by step:

Step 1: Determine the First Digit

The first digit of any 4-digit number should be greater than 3 to ensure the number is higher than 4000. Therefore, the first digit can be 4, 5, or 6. This gives us 3 possible choices:

4 5 6

Step 2: Determine the Remaining Digits

For the remaining three digits (second, third, and fourth), any of the available digits (0, 2, 3, 4, 5, and 6) can be used. Since repetition is allowed, each of these positions can be filled with any of the 6 digits:

Second digit: 6 choices Third digit: 6 choices Fourth digit: 6 choices

Step 3: Calculate the Total Combinations

To find the total number of 4-digit numbers that meet the criteria, we multiply the number of choices for each digit position:

begin{equation}text{Total Combinations} text{Choices for first digit} times text{Choices for second digit} times text{Choices for third digit} times text{Choices for fourth digit}end{equation}

Substituting the values, we get:

begin{equation}text{Total Combinations} 3 times 6 times 6 times 6 648end{equation}

Conclusion: Therefore, 648 different 4-digit numbers greater than 4000 can be formed using the digits 0, 2, 3, 4, 5, and 6 with repetition allowed.

Additional Consideration for Even Numbers

For each 4-digit even number formed, the following constraints apply:

The first digit can still be 4, 5, or 6 (3 choices) The last digit must be 0, 2, 4, or 6 (4 choices) The second and third digits can be any of the 7 available digits (0, 1, 2, 3, 4, 5, 6) (7 choices each)

Using these constraints, the formula to calculate the number of 4-digit even numbers is:

begin{equation}text{Total Combinations} 3 times 7 times 7 times 4 588end{equation}

Note: To exclude the number 4000, we subtract 1 from the total, resulting in:

begin{equation}588 - 1 587end{equation}

Hence, there are 587 4-digit even numbers greater than 4000 that can be formed using the digits 0, 1, 2, 3, 4, 5, and 6, with repetition allowed.