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

Options

A. 72
B. 144
C. 288
D. 720

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, where the digits do not repeat and the terminal digits are even, we can follow these steps:

### Step 1: Identify the even digits
The even digits available from the set {1, 2, 3, 4, 5, 6, 7} are 2, 4, and 6. Thus, we have 3 choices for the terminal digits.

**Hint:** Remember that terminal digits refer to the first and last digits of the number.

### Step 2: Choose the terminal digits
We can choose one of the even digits for the first position (let's denote it as the first terminal digit) and one of the remaining even digits for the last position (the second terminal digit).

- If we choose one even digit for the first position, we have 2 remaining choices for the last position.

**Hint:** The first and last digits must be different and even.

### Step 3: Count the remaining digits
After fixing the two terminal digits, we have 5 digits left to choose from (the remaining digits from the original set minus the two chosen even digits). These remaining digits include 1, 3, 5, 7, and the unused even digit.

**Hint:** Always keep track of the digits that have already been used.

### Step 4: Fill the middle four positions
Now, we need to fill the middle four positions with the remaining 5 digits. The number of ways to choose and arrange 4 digits from these 5 remaining digits is calculated as follows:

- For the first middle position, we have 5 choices.
- For the second middle position, we have 4 choices (one digit is used).
- For the third middle position, we have 3 choices.
- For the fourth middle position, we have 2 choices.

**Hint:** The number of choices decreases as we fill each position.

### Step 5: Calculate the total combinations
Now we can calculate the total number of combinations:

1. Choose the first terminal digit: 3 choices (2, 4, or 6).
2. Choose the last terminal digit: 2 choices (the remaining even digits).
3. Choose and arrange the middle four digits: \(5 \times 4 \times 3 \times 2\).

Putting it all together, the total number of six-digit numbers is:

\[ \text{Total} = 3 \times 2 \times (5 \times 4 \times 3 \times 2) \]

Calculating this:

\[ = 3 \times 2 \times 120 \]
\[ = 6 \times 120 = 720 \]

### Final Answer
The total number of six-digit numbers that can be formed under the given conditions is **720**.

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