How many 4 digit numbers are there, without repetition of digits, If each number is divisible by 5 ?

Answer

To solve the problem of finding how many 4-digit numbers can be formed without repetition of digits, where each number is divisible by 5, we can break it down into two cases based on the last digit.

### Step 1: Understand the conditions
A 4-digit number must be divisible by 5, which means its last digit must be either 0 or 5.

### Step 2: Case 1 - Last digit is 0
1. **Fix the last digit**: The last digit is 0.
2. **Choose the first digit**: The first digit can be any digit from 1 to 9 (since it cannot be 0). This gives us 9 options.
3. **Choose the second digit**: After selecting the first digit, we have 8 remaining digits to choose from (0 is already used as the last digit).
4. **Choose the third digit**: After selecting the first and second digits, we have 7 remaining digits to choose from.

The total number of combinations for this case is calculated as follows:
- First digit: 9 options (1-9)
- Second digit: 8 options (remaining digits)
- Third digit: 7 options (remaining digits)

So, the total for this case is:
\[ 9 \times 8 \times 7 = 504 \]

### Step 3: Case 2 - Last digit is 5
1. **Fix the last digit**: The last digit is 5.
2. **Choose the first digit**: The first digit can be any digit from 1 to 9 (it cannot be 0 or 5). This gives us 8 options.
3. **Choose the second digit**: After selecting the first digit, we can choose from the remaining digits (including 0 but excluding the first digit and 5). This gives us 8 options.
4. **Choose the third digit**: After selecting the first and second digits, we have 7 remaining digits to choose from.

The total number of combinations for this case is calculated as follows:
- First digit: 8 options (1-9 excluding 5)
- Second digit: 8 options (remaining digits including 0)
- Third digit: 7 options (remaining digits)

So, the total for this case is:
\[ 8 \times 8 \times 7 = 448 \]

### Step 4: Combine both cases
Now we add the totals from both cases to find the overall total number of 4-digit numbers divisible by 5:
\[ 504 + 448 = 952 \]

### Final Answer
Thus, the total number of 4-digit numbers without repetition of digits that are divisible by 5 is **952**.

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