Solution

The correct answer is 86,658 

Key PointsWe want to find the sum of all four-digit numbers that can be formed using the digits 1, 2, 4, and 6. Each distinct digit will occupy each place value (Thousands, Hundreds, Tens, and Units) the same number of times when we consider all possible permutations.

There are total 4! = 4*3*2*1 = 24 ways to permute 4 different digits in 4 places.

The sum of all 4-digit numbers can be found by calculating the sum for each of the 4 positions (Thousands, Hundreds, Tens, and Units), then summing those results.

• For each position:
  • Each of the 4 numbers (1, 2, 4, 6) will appear in each position 1/4th of the time in the total permutations, so 24 / 4 = 6 times for each.
  • The sum of the digits is 1 + 2 + 4 + 6 = 13.
  • So, the contribution for each position will be 13 * 6 = 78.
• Now, we calculate the total sum taking into account the place value:
  • The Thousands place contributes 78 * 1000 = 78,000.
  • The Hundreds place contribute 78 * 100 = 7,800.
  • The Tens place gives 78 * 10 = 780.
  • The Units place contributes 78 * 1 = 78.

Adding those up, the total sum of all 4-digit numbers that can be made with the digits 1, 2, 4, and 6 is 78,000 + 7,800 + 780 + 78 = 86,658. So the answer is option 1) 86,658.