How many numbers divisible by 125 can be written by using all the digits 2,3,8,7 and 5 exactly once?

Options

A. 4
B. 5
C. 6
D. 7

Answer

To determine how many numbers divisible by 125 can be formed using the digits 2, 3, 8, 7, and 5 exactly once, we need to follow these steps:

### Step 1: Identify the condition for divisibility by 125
A number is divisible by 125 if its last three digits form a number that is a multiple of 125. The multiples of 125 that can be formed with the digits available (2, 3, 8, 7, 5) must be considered.

### Step 2: List the relevant multiples of 125
The relevant multiples of 125 that can be formed with the digits we have are:
- 375
- 875

### Step 3: Check which multiples can be formed
Now, we check if we can form the numbers 375 and 875 using the digits 2, 3, 8, 7, and 5.

1. **For 375**:
- The digits used are 3, 7, and 5.
- The remaining digits are 2 and 8.
- Thus, we can form the number 375 with the remaining digits in front:
- Possible combinations: 28375, 82375, 28735, 82735, 28573, 82573, etc.
- The total arrangements of the remaining digits (2 and 8) in front of 375 is 2! = 2.

2. **For 875**:
- The digits used are 8, 7, and 5.
- The remaining digits are 2 and 3.
- Thus, we can form the number 875 with the remaining digits in front:
- Possible combinations: 23875, 32875, 23587, 32587, etc.
- The total arrangements of the remaining digits (2 and 3) in front of 875 is also 2! = 2.

### Step 4: Calculate the total numbers formed
Now, we add the total combinations from both cases:
- From 375: 2 combinations
- From 875: 2 combinations

Total combinations = 2 + 2 = 4.

### Final Answer
Thus, the total number of numbers divisible by 125 that can be formed using the digits 2, 3, 8, 7, and 5 exactly once is **4**.