Final answer:
The average of all 5-digit numbers formed with the digits 1, 3, 5, 7, 9 is 13,579. Each digit contributes equally to each position in the number, resulting in an average contribution of 2,715.8 per position, which is then multiplied by 5 to find the overall average.
Step-by-step explanation:
To calculate the average of all 5-digit numbers that can be formed using the digits 1, 3, 5, 7, 9, we must understand that each digit will appear in each position (tens of thousands, thousands, hundreds, tens, ones) an equal number of times when we list all possible permutations of these digits. Since there are 5 different positions and each number can only appear in one position at a time in a number, each digit will appear in each position ⅔ (1/5) of the time.
For a 5-digit number, the value contributed by a digit in the tens of thousands, thousands, hundreds, tens, and ones positions can be calculated by multiplying the digit by 10,000, 1,000, 100, 10, and 1, respectively. Therefore, the average value contributed by each digit in its respective position to the overall average is:
Average contribution per position = (1 x 10,000 + 3 x 1,000 + 5 x 100 + 7 x 10 + 9 x 1) / 5 = (10,000 + 3,000 + 500 + 70 + 9) / 5 = 13,579 / 5 = 2,715.8
Since each position has the same average contribution and there are 5 positions, the overall average is:
Overall average = 2,715.8 x 5 = 13,579
Therefore, the average of all 5-digit numbers formed with the digits 1, 3, 5, 7, 9 is 13,579.