Question:

The available digits are $\{0, 1, 3, 7, 9\}$. Repetition of digits is allowed. We seek to form natural numbers less than 7,000. Based on the context, this refers specifically to 4-digit numbers. **Step 1: Determine the constraints for a 4-digit number less than 7,000.** Let the 4-digit number be represented as $ABCD$, where $A$ is the thousands digit, $B$ is the hundreds digit, $C$ is the tens digit, and $D$ is the units digit. For a number to be a 4-digit number, the thousands digit $A$ cannot be $0$. For the number to be less than 7,000, the thousands digit $A$ must be less than $7$. Considering the available digits $\{0, 1, 3, 7, 9\}$: * The thousands digit $A$ can be $1$ or $3$. (It cannot be $0$, $7$, or $9$). This gives 2 choices for $A$. * The hundreds digit $B$, tens digit $C$, and units digit $D$ can be any of the available digits $\{0, 1, 3, 7, 9\}$, as repetition is allowed. This gives 5 choices for each of $B$, $C$, and $D$. **Step 2: Calculate the total number of such 4-digit numbers.** The number of choices for each position are: * Thousands digit ($A$): 2 choices ($\{1, 3\}$) * Hundreds digit ($B$): 5 choices ($\{0, 1, 3, 7, 9\}$) * Tens digit ($C$): 5 choices ($\{0, 1, 3, 7, 9\}$) * Units digit ($D$): 5 choices ($\{0, 1, 3, 7, 9\}$) The total number of 4-digit natural numbers less than 7,000 that can be formed is the product of the number of choices for each digit: $$2 \times 5 \times 5 \times 5 = 250$$