How many numbers between 6000 and 7000 a divisible by 5 which can be formed with the digits 5,6,7 and 9?

Answer

To solve the problem of how many numbers between 6000 and 7000 that are divisible by 5 can be formed using the digits 5, 6, 7, and 9, we can follow these steps:

### Step 1: Understand the constraints
We need to form a four-digit number between 6000 and 7000. This means the first digit must be 6. Additionally, since the number must be divisible by 5, the last digit must be 5.

### Step 2: Fix the first and last digits
Based on the constraints:
- The first digit (thousands place) is fixed as 6.
- The last digit (units place) is fixed as 5.

So, our number looks like this: **6 _ _ 5**

### Step 3: Identify the remaining digits
After fixing 6 and 5, we have the remaining digits 7 and 9 available for the two middle places.

### Step 4: Arrange the remaining digits
We need to fill the two middle places with the digits 7 and 9. The arrangement of two digits can be calculated using the factorial of the number of digits:
- The number of arrangements of 7 and 9 in the two middle positions is given by \(2!\) (2 factorial).

Calculating \(2!\):
\[ 2! = 2 \times 1 = 2 \]

### Step 5: List the possible numbers
The two arrangements of the digits 7 and 9 in the middle positions are:
1. 6795
2. 6975

### Conclusion
Thus, the total number of four-digit numbers between 6000 and 7000 that are divisible by 5 and can be formed using the digits 5, 6, 7, and 9 is **2**.