The probability of 5 digit numbers that are made up of exactly two distinct digit is

Options

A. `135/10^4`
B. `125,10^4`
C. `144/10^4`
D. `127/10^4`

Answer

To find the probability of 5-digit numbers that are made up of exactly two distinct digits, we will follow these steps:

### Step 1: Calculate the total number of 5-digit numbers
The range of 5-digit numbers is from 10,000 to 99,999. Therefore, the total number of 5-digit numbers can be calculated as:

\[ \text{Total 5-digit numbers} = 99,999 - 10,000 + 1 = 90,000 \]

### Step 2: Choose two distinct digits
We need to choose 2 distinct digits from the digits 0 to 9. However, since we are forming a 5-digit number, we cannot have 0 as the leading digit. Thus, we can choose 2 digits from the non-zero digits (1 to 9) and include 0 as one of the digits if needed.

The number of ways to choose 2 distinct digits from the 9 non-zero digits is given by:

\[ \binom{9}{2} \]

### Step 3: Count the arrangements of the digits
Once we have chosen 2 distinct digits, we need to consider the different ways to arrange these digits in a 5-digit number. The arrangements can be classified based on how many times each digit appears.

1. **Case 1**: One digit appears 4 times, and the other appears 1 time.
- There are 2 ways to choose which digit appears 4 times.
- The number of arrangements is given by:
\[ \frac{5!}{4! \cdot 1!} = 5 \]

2. **Case 2**: One digit appears 3 times, and the other appears 2 times.
- There are 2 ways to choose which digit appears 3 times.
- The number of arrangements is given by:
\[ \frac{5!}{3! \cdot 2!} = 10 \]

3. **Case 3**: One digit appears 2 times, and the other appears 3 times.
- This is similar to Case 2 and gives:
\[ \frac{5!}{2! \cdot 3!} = 10 \]

### Step 4: Total arrangements for a chosen pair of digits
The total arrangements for a chosen pair of digits can be calculated by summing the contributions from all cases:

\[ \text{Total arrangements} = 5 + 10 + 10 = 25 \]

### Step 5: Total valid 5-digit numbers with exactly two distinct digits
Now, we multiply the number of ways to choose the digits by the total arrangements:

\[ \text{Total valid numbers} = \binom{9}{2} \times 25 \]

Calculating \(\binom{9}{2}\):

\[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]

Thus, the total valid 5-digit numbers is:

\[ \text{Total valid numbers} = 36 \times 25 = 900 \]

### Step 6: Calculate the probability
Finally, the probability of randomly selecting a 5-digit number that is made up of exactly two distinct digits is given by:

\[ \text{Probability} = \frac{\text{Total valid numbers}}{\text{Total 5-digit numbers}} = \frac{900}{90,000} = \frac{1}{100} \]

### Final Answer
The probability of 5-digit numbers that are made up of exactly two distinct digits is:

\[ \frac{1}{100} \]