How many 6 digit numbers can be formed out of the number 567724, which are even?

Options

A. a. 360
B. b. 480
C. c. 180
D. d. 220

Answer

To find how many 6-digit even numbers can be formed from the digits of the number 567724, we need to consider the even digits available and the arrangements of the remaining digits. The even digits in 567724 are 2, 4, and 6.

### Step-by-Step Solution:

1. **Identify the even digits**: The even digits in the number 567724 are 2, 4, and 6. Since we want to form even numbers, the last digit of our 6-digit number must be one of these even digits.

2. **Case 1: Last digit is 2**:
- If we fix the last digit as 2, the remaining digits are 5, 6, 7, 7, and 4.
- The number of arrangements of these digits is calculated as:
\[ \text{Arrangements} = \frac{5!}{2!} = \frac{120}{2} = 60 \]
- (We divide by \(2!\) because the digit 7 is repeated twice.)

3. **Case 2: Last digit is 4**:
- If we fix the last digit as 4, the remaining digits are 5, 6, 7, 7, and 2.
- The number of arrangements is again:
\[ \text{Arrangements} = \frac{5!}{2!} = \frac{120}{2} = 60 \]

4. **Case 3: Last digit is 6**:
- If we fix the last digit as 6, the remaining digits are 5, 7, 7, 2, and 4.
- The number of arrangements is:
\[ \text{Arrangements} = \frac{5!}{2!} = \frac{120}{2} = 60 \]

5. **Total arrangements**:
- Now, we sum the arrangements from all three cases:
\[ \text{Total} = 60 + 60 + 60 = 180 \]

Thus, the total number of 6-digit even numbers that can be formed from the digits of 567724 is **180**.