Mr . A gave his telephone number to Mr . B .Mr . B remembers that the first two digits were 40 and the remaining four digits were two 3s, one 6 and one 8 . He is not certain about the order of the digits . Mr. B dials 403638. The probability that he will get A ' s house is

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(1)/(8)`
D. `(1)/(12)`

Answer

To solve the problem, we need to determine the probability that Mr. B dials the correct telephone number of Mr. A, given the information he remembers.

### Step-by-Step Solution:

1. **Identify the Known Information:**
- The first two digits of the telephone number are 40.
- The remaining four digits consist of two 3s, one 6, and one 8.

2. **List the Remaining Digits:**
- The remaining digits are: 3, 3, 6, 8.

3. **Calculate the Total Arrangements of the Remaining Digits:**
- Since there are 4 digits where '3' is repeated twice, we can use the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{n!}{n_1! \cdot n_2! \cdots n_k!} \]
where \( n \) is the total number of items to arrange, and \( n_1, n_2, \ldots, n_k \) are the counts of each distinct item.
- Here, \( n = 4 \) (total digits), \( n_1 = 2 \) (for the two 3s), and \( n_2 = 1 \) (for 6), \( n_3 = 1 \) (for 8).
- Thus, the total arrangements can be calculated as:
\[ \text{Total arrangements} = \frac{4!}{2! \cdot 1! \cdot 1!} = \frac{24}{2} = 12. \]

4. **Determine the Number of Successful Outcomes:**
- There is only 1 specific arrangement that corresponds to Mr. A's telephone number, which is 403638.

5. **Calculate the Probability:**
- The probability \( P \) that Mr. B dials the correct number is given by the ratio of successful outcomes to total outcomes:
\[ P = \frac{\text{Number of successful outcomes}}{\text{Total arrangements}} = \frac{1}{12}. \]

### Final Answer:
The probability that Mr. B will get Mr. A's house is \( \frac{1}{12} \).

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