To solve the problem of finding the probability of forming a 5-digit number with specific conditions (2 in the tens place and 0 in the units place), we'll break it down into steps.

Step 1: Total Number of 5-Digit Numbers

Given that we can use the digits $0, 1, 2, 3, 4$, let's find the total number of 5-digit numbers that can be formed:

• The first digit (the thousands place) cannot be $0$ (since it's a 5-digit number), so it has 4 possible choices: $1, 2, 3, 4$.
• The second digit has 5 possible choices: $0, 1, 2, 3, 4$.
• The third digit also has 5 possible choices: $0, 1, 2, 3, 4$.
• The fourth digit also has 5 possible choices: $0, 1, 2, 3, 4$.
• The fifth digit (the units place) also has 5 possible choices: $0, 1, 2, 3, 4$.

So, the total number of possible 5-digit numbers is: $4 \times 5 \times 5 \times 5 \times 5 = 2500$

Step 2: Number of Favorable Outcomes

Now, we need to count the favorable outcomes where the number has $2$ in the tens place and $0$ in the units place.

• The fifth digit (units place) is fixed as $0$.
• The fourth digit (tens place) is fixed as $2$.
• The first digit (thousands place) has 4 possible choices: $1, 2, 3, 4$.
• The second digit has 5 possible choices: $0, 1, 2, 3, 4$.
• The third digit also has 5 possible choices: $0, 1, 2, 3, 4$.

So, the number of favorable outcomes is: $4 \times 5 \times 5 = 100$

Step 3: Calculate the Probability

The probability is given by the ratio of the number of favorable outcomes to the total number of possible outcomes: $\text{Probability} = \frac{100}{2500} = \frac{1}{25}$

Final Answer

The probability of getting $2$ in the tens place and $0$ in the units place is $\boxed{\frac{1}{25}}$.

Would you like to know more details or have any questions?

Here are some related questions:

1. What is the probability of getting the digit $4$ in the thousands place?
2. How many different 5-digit numbers can be formed if the digits cannot repeat?
3. What is the probability of forming a number with at least one digit as $3$?
4. How does the probability change if the digits can repeat, except $0$ cannot be used?
5. What is the probability that all digits in the number are even?

Tip: Always double-check your conditions, such as restrictions on digits, to ensure accuracy when calculating probabilities.