Find the sum of all numbers greater than 10000 formed by using the digits 1,3,5,7,9 no digit being repeated in any number

To find the sum of all numbers greater than 10,000 that can be formed using the digits 1, 3, 5, 7, and 9, where no digit repeats, we need to consider the different combinations of these digits that can create valid numbers. Since we are limited to five unique digits, all valid numbers will be five digits long, as any four-digit numbers will not meet the requirement of being greater than 10,000.

Step 1: Understanding the digit placement

Since the numbers must be five digits, we can use all five available digits. The first digit must be greater than 0, and since we are only using the digits 1, 3, 5, 7, and 9, any of these can be the first digit. This means all five digits can serve as the leading digit.

Step 2: Calculating the number of permutations

To find out how many five-digit numbers can be formed, we can calculate the permutations of the five digits. The formula for permutations of n unique items is n! (n factorial). For our case:

• n = 5 (the digits 1, 3, 5, 7, 9)
• Calculating 5! = 5 × 4 × 3 × 2 × 1 = 120

There are 120 unique five-digit numbers that can be formed using these digits.

Step 3: Summing the values of each digit's contribution

Next, we need to calculate the sum of all these numbers. A systematic way to do this is to determine how many times each digit appears in each place value (ten-thousands, thousands, hundreds, tens, and units).

Digit Frequency in Each Position

Each digit will appear in each position an equal number of times. Since there are 120 total numbers and 5 possible leading digits, each digit will occupy each position the following number of times:

• Each digit appears in each position: 120 total numbers / 5 digits = 24 times.

Calculating the contribution of each position

Now, we can calculate the contribution of each digit to the total sum based on its position:

• Ten-thousands place contributes: 24 × 10,000 × (1 + 3 + 5 + 7 + 9)
• Thousands place contributes: 24 × 1,000 × (1 + 3 + 5 + 7 + 9)
• Hundreds place contributes: 24 × 100 × (1 + 3 + 5 + 7 + 9)
• Tens place contributes: 24 × 10 × (1 + 3 + 5 + 7 + 9)
• Units place contributes: 24 × 1 × (1 + 3 + 5 + 7 + 9)

First, let's calculate the sum of the digits:

1 + 3 + 5 + 7 + 9 = 25

Final Calculation

Now we can plug this sum into our contributions:

• Ten-thousands: 24 × 10,000 × 25 = 6,000,000
• Thousands: 24 × 1,000 × 25 = 600,000
• Hundreds: 24 × 100 × 25 = 60,000
• Tens: 24 × 10 × 25 = 6,000
• Units: 24 × 1 × 25 = 600

Adding these contributions together gives:

6,000,000 + 600,000 + 60,000 + 6,000 + 600 = 6,666,600

Thus, the sum of all numbers greater than 10,000 that can be formed using the digits 1, 3, 5, 7, and 9, without repeating any digit, is 6,666,600.