Using the digits 1,5,2,8 four digit numbers are formed and the sum of all possible such numbers.Correct answer is '106656'. Can you explain this answer?

Question

Answer

Explanation:

To form all possible four-digit numbers using the digits 1, 5, 2, and 8, we need to use each of the digits once in each of the four positions. This gives us a total of 4! = 24 different numbers.

Calculating the sum:

To calculate the sum of all possible four-digit numbers, we need to find the sum of the digits in each position. Since each digit appears in each position equally often, we can simply add up the digits in each position and multiply by the number of numbers.

Thousands place:

The digits that can go in the thousands place are 1, 5, 2, and 8. The sum of these digits is 1 + 5 + 2 + 8 = 16.

To calculate the total sum of all possible numbers, we need to multiply this sum by the number of numbers that can be formed using these digits in the thousands place. Since each of the remaining digits can be used in the hundreds, tens, and ones places, there are 3! = 6 possible numbers that can be formed using these digits. Therefore, the sum of the digits in the thousands place contributes 16 x 6 x 1000 = 96000 to the total sum.

Hundreds place:

The digits that can go in the hundreds place are 1, 5, 2, and 8 (excluding the digit used in the thousands place). The sum of these digits is 1 + 5 + 2 + 8 = 16.

To calculate the total sum of all possible numbers, we need to multiply this sum by the number of numbers that can be formed using these digits in the hundreds place. Since each of the remaining digits can be used in the tens and ones places, there are 2! = 2 possible numbers that can be formed using these digits. Therefore, the sum of the digits in the hundreds place contributes 16 x 2 x 100 = 3200 to the total sum.

Tens place:

The digits that can go in the tens place are 1, 5, 2, and 8 (excluding the digits used in the thousands and hundreds places). The sum of these digits is 1 + 5 + 2 + 8 = 16.

To calculate the total sum of all possible numbers, we need to multiply this sum by the number of numbers that can be formed using these digits in the tens place. Since there is only one digit left to be used in the ones place, there is only 1 possible number that can be formed using these digits. Therefore, the sum of the digits in the tens place contributes 16 x 1 x 10 = 160 to the total sum.

Ones place:

The digits that can go in the ones place are 1, 5, 2, and 8 (excluding the digits used in the thousands, hundreds, and tens places). The sum of these digits is 1 + 5 + 2 + 8 = 16.

To calculate the total sum of all possible numbers, we need to multiply this sum by the number of numbers that can be formed using these digits in the ones place

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