A person while dialing a telephone number, forgets the last 3 digits of the number but remembers that exactly two of them are same. He dials the number randomly. What is the probability that he dialed the correct number.

To solve this problem, we need to determine the probability that the person correctly dials the last three digits of the telephone number, given that they know two of the digits are the same.

### Step 1: Determine the Total Number of Possible Combinations
Since the last three digits of the telephone number are involved, and exactly two of them are the same, we have a few scenarios:

Let's denote the three digits as \( a \), \( a \), and \( b \), where \( a \) and \( b \) can be any digit from 0 to 9, but \( a \neq b \).

There are 10 possible choices for \( a \) (from 0 to 9) and 9 possible choices for \( b \) (since \( b \) must be different from \( a \)).

Next, we need to account for the different arrangements of these digits. The digits can be arranged in the following ways:

- \( aab \)
- \( aba \)
- \( baa \)

Thus, for each pair \( (a, b) \), there are 3 possible arrangements.

So, the total number of possible combinations of three digits where exactly two digits are the same is:

\[
\text{Total combinations} = 10 \times 9 \times 3 = 270
\]

### Step 2: Determine the Number of Favorable Combinations
Since the person is dialing the number randomly, and there is only one correct combination out of the 270 possible combinations, the number of favorable outcomes is 1.

### Step 3: Calculate the Probability
The probability that the person correctly dials the last three digits is the ratio of the number of favorable outcomes to the total number of possible outcomes:

\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible combinations}} = \frac{1}{270}
\]

### Final Answer
The probability that the person dials the correct number is \( \frac{1}{270} \).

Would you like to explore this problem further or ask any questions about the steps?

Here are five questions related to this problem:
1. How would the probability change if the person remembered that all three digits were different?
2. What if the person remembered that all three digits were the same? What would be the probability then?
3. How can the concept of permutations help in solving similar probability problems?
4. What would be the total number of combinations if the person didn't remember any restrictions on the digits?
5. How does the probability change if the number of digits forgotten increases?

**Tip:** When calculating probabilities, always ensure you carefully distinguish between permutations (arrangements of items) and combinations (selections of items).

Explore the probability of dialing the correct telephone number when the last three digits are forgotten but two are known to be the same. Learn the step-by-step solution, including total and favorable combinations, and calculate the probability involved. Suitable for high school students.

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