Animated Solution for Mathematics - Permutations and Combinations: The number of natural numbers less than 7,000 which can be formed by using the digits 0,1,3,7,9 (repetition of digits allowed) is equal to :

To determine how many natural numbers less than $7,000$ can be formed using the set of digits $\{0,1,3,7,9\}$, we must categorize the numbers by their digit count. Since the numbers must be natural, we exclude $0$ as a standalone number.

A number less than $7,000$ can consist of $1,2,3,$ or $4$ digits. We will calculate the possibilities for each case independently.

For 1-digit numbers, the valid choices are $\{1,3,7,9\}$. This gives us $4$ possibilities.

For 2-digit numbers, the tens place cannot be $0$. Thus, we have $4$ choices for the tens place $\{1,3,7,9\}$ and $5$ choices for the units place $\{0,1,3,7,9\}$.

For 3-digit numbers, the hundreds place cannot be $0$. We have $4$ choices for the hundreds place and $5$ choices for both the tens and units places.

To ensure the 4-digit number is less than $7,000$, the thousands digit must be restricted. The thousands digit cannot be $0$ (as that would result in a 3-digit number), and it cannot be $7$ or $9$ (as that would result in a number $\ge 7,000$).

Therefore, the thousands place can only be $1$ or $3$, providing $2$ choices. The remaining three positions (hundreds, tens, and units) can each be any of the $5$ available digits.

Since these cases are mutually exclusive, we find the total count by summing the results of each phase:

The total number of natural numbers less than $7,000$ that can be formed using the given digits is 374.