The number of six digit numbers that can be formed from the digits 1,2,3,4,5,6 and 7 so that digits do not repeat and the terminal digits are even, is

Options

A. A. 388
B. B. 555
C. C.720
D. D. none of these

Answer

To solve the problem of finding the number of six-digit numbers that can be formed from the digits 1, 2, 3, 4, 5, 6, and 7, with the condition that the digits do not repeat and the terminal digits are even, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the even digits**: The even digits available from the set {1, 2, 3, 4, 5, 6, 7} are 2, 4, and 6. These digits will occupy the terminal positions (first and last) of the six-digit number.

2. **Choose the last digit**: Since the last digit must be even, we have 3 options (2, 4, or 6) for the last digit.

3. **Choose the first digit**: After choosing the last digit, we have 2 remaining even digits left. Therefore, we have 2 options for the first digit.

4. **Choose the middle four digits**: After selecting the first and last digits, we will have used 2 digits from the original set of 7. This leaves us with 5 digits remaining (7 total - 2 used = 5 remaining). We need to choose 4 digits from these 5 remaining digits.

5. **Calculate the arrangements of the middle four digits**: The 4 digits can be arranged in 4! (factorial) ways.

6. **Combine all choices**: The total number of six-digit numbers can be calculated by multiplying the number of choices for each step:
- Choices for the last digit: 3
- Choices for the first digit: 2
- Arrangements of the middle four digits: 4! = 24

### Final Calculation:
The total number of six-digit numbers is given by:
\[ \text{Total} = (\text{Choices for last digit}) \times (\text{Choices for first digit}) \times (\text{Arrangements of middle digits}) \]
\[ \text{Total} = 3 \times 2 \times 24 = 144 \]

### Conclusion:
Thus, the total number of six-digit numbers that can be formed under the given conditions is **144**.

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