An old man while dialing a 7-digit telephone number remembers that the first four digits consist of one 1's, one 2's and two 3's. he also remembers that the fifth digit is either a 4 or 5 while has no memory of the sixt digit, he remembers that the seventh digit is 9 minus the sixth digit, maximum number of distinct trials he has to try to make sure that he dials the correct telephone number, is

A. 360
B. 240
C. 216
D. none of these

Options

A. 360
B. 240
C. 216
D. none of these

Answer

To solve the problem step by step, we need to find the maximum number of distinct trials the old man has to make to dial the correct telephone number based on the information he remembers.

### Step 1: Arranging the First Four Digits
The first four digits consist of one 1, one 2, and two 3's. To find the number of distinct arrangements of these digits, we can use the formula for permutations of multiset:

\[ \text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times n_3!} \]

where \( n \) is the total number of items to arrange, and \( n_1, n_2, n_3 \) are the counts of each distinct item.

Here, we have:
- Total digits \( n = 4 \) (1, 2, 3, 3)
- Counts: 1 of '1', 1 of '2', and 2 of '3'.

Thus, the calculation is:

\[ \text{Number of arrangements} = \frac{4!}{1! \times 1! \times 2!} = \frac{24}{1 \times 1 \times 2} = \frac{24}{2} = 12 \]

### Step 2: Choosing the Fifth Digit
The fifth digit can either be a 4 or a 5. This gives us 2 options for the fifth digit.

### Step 3: Determining the Sixth and Seventh Digits
The sixth digit is unknown, so we denote it as \( x \). The seventh digit is given as \( 9 - x \).

The possible values for \( x \) (the sixth digit) can be any digit from 0 to 9. However, since the seventh digit must also be a valid digit, we need to ensure that \( 9 - x \) is also a digit (0-9).

This gives us the following valid pairs for \( (x, 9 - x) \):
- \( (0, 9) \)
- \( (1, 8) \)
- \( (2, 7) \)
- \( (3, 6) \)
- \( (4, 5) \)
- \( (5, 4) \)
- \( (6, 3) \)
- \( (7, 2) \)
- \( (8, 1) \)
- \( (9, 0) \)

Thus, there are 10 valid combinations for the sixth and seventh digits.

### Step 4: Calculating the Total Combinations
Now we can combine all the possibilities:

\[ \text{Total combinations} = (\text{arrangements of first 4 digits}) \times (\text{choices for 5th digit}) \times (\text{combinations for 6th and 7th digits}) \]

Substituting the values we found:

\[ \text{Total combinations} = 12 \times 2 \times 10 = 240 \]

### Final Answer
The maximum number of distinct trials the old man has to try to ensure he dials the correct telephone number is **240**.