Answer :
There are a total of 655 natural numbers less than 7,000 that can be formed using the digits 0, 1, 3, 7, 9, with digit repetition allowed. None of the given option is correct and the correct answer is 655.
The student's question asks how many natural numbers less than 7,000 can be formed using the digits 0, 1, 3, 7, 9 with repetition allowed. To find the answer, we must consider four-digit numbers where the first digit cannot be 0 (to ensure we have a number less than 7000), and all other digits can be any of the five given numbers.
For the first digit, there are 4 possibilities (1, 3, 7, or 9), since 0 cannot be used. For each of the remaining three places, we have 5 choices since repetition is allowed and we can use 0 here. So we have 4 x 5 x 5 x 5 = 500 possible numbers, and this does not include numbers with fewer digits.
For one, two, or three-digit numbers, since we can use the leading digit as 0, there are 5 possibilities for each place, giving us 5 + 5 x 5 + 5 x 5 x 5 = 5 + 25 + 125 = 155 options.
Add the two results together for the total number of possibilities: 500 + 155 = 655.
