Introduction

Finding the sum of all 4-digit numbers using specific digits can be approached systematically by considering the placement of each digit in the number. This article will guide you through the process, breaking down the steps to find the sum of all 4-digit numbers that can be formed using the digits 0, 1, 6, 7, and 8.

Digit Selection and Placement Constraints

To form a valid 4-digit number using the digits 0, 1, 6, 7, and 8, the first digit cannot be 0. This means the possible choices for the first digit are 1, 6, 7, and 8. Each of the remaining three digits can be any of the five digits (0, 1, 6, 7, 8), allowing for repetition.

Counting Valid 4-Digit Numbers

Let's begin by counting all possible 4-digit numbers that can be formed under these constraints.

For the first digit, there are 4 possible choices (1, 6, 7, 8).

For the second, third, and fourth digits, there are 5 possible choices each (0, 1, 6, 7, 8).

The total number of 4-digit numbers is:

t4 (first digit choices) × 5 (second digit choices) × 5 (third digit choices) × 5 (fourth digit choices) 4 × 125 500

Calculating the Contribution of Each Digit Position

To calculate the sum, we need to consider the contribution of each digit in each position (thousands, hundreds, tens, and units).

Contribution of Each Position

Thousands Place

Each of the digits 1, 6, 7, and 8 can be in the thousands place. There are 125 combinations for each digit (5 choices for each of the other three digits).

The total contribution from the thousands place is:

t(1 6 7 8) × 125 × 1000 22 × 125 × 1000 2,750,000

Hundreds Place

Any of the 5 digits (0, 1, 6, 7, 8) can be in the hundreds place. Each digit will appear in the hundreds place 100 times (4 choices for the first digit and 5 choices for each of the other two digits).

The total contribution from the hundreds place is:

t(0 1 6 7 8) × 100 × 100 22 × 100 × 100 220,000

Tens Place

The same logic applies to the tens place. Each digit will appear 100 times in the tens place.

The total contribution from the tens place is:

t(0 1 6 7 8) × 100 × 10 22 × 100 × 10 22,000

Units Place

Each digit will appear 100 times in the units place.

The total contribution from the units place is:

t(0 1 6 7 8) × 100 × 1 22 × 100 2,200

Total Sum Calculation

Finally, we sum the contributions from all digit places to find the total sum:

tTotal Sum 2,750,000 220,000 22,000 2,200 2,994,200

Conclusion

The sum of all 4-digit numbers that can be formed using the digits 0, 1, 6, 7, and 8 is 2,994,200. By understanding the placement and contribution of each digit, we can systematically solve complex digit-based problems.