As is the small four digit number formed by using all the digits 0,1,2, and 3. B is the greatest four digit number formed by using all the digits 0, 1,2 and 3. Find B-A.

Answer

To solve the problem, we need to find the smallest four-digit number \( A \) and the greatest four-digit number \( B \) that can be formed using the digits 0, 1, 2, and 3. Then, we will calculate \( B - A \).

### Step 1: Find the smallest four-digit number \( A \)

1. The smallest four-digit number cannot start with 0, as that would make it a three-digit number. Therefore, we will start with the smallest digit available that is not 0, which is 1.
2. The remaining digits are 0, 2, and 3. We will arrange these digits in ascending order after 1.
3. Thus, the smallest four-digit number \( A \) is formed as follows:
- Start with 1
- Next, use 0, then 2, and finally 3
- Therefore, \( A = 1023 \)

### Step 2: Find the greatest four-digit number \( B \)

1. To find the greatest four-digit number, we will arrange the digits in descending order.
2. The digits available are 0, 1, 2, and 3. The largest digit is 3, followed by 2, then 1, and finally 0.
3. Thus, the greatest four-digit number \( B \) is formed as follows:
- Start with 3
- Next, use 2, then 1, and finally 0
- Therefore, \( B = 3210 \)

### Step 3: Calculate \( B - A \)

1. Now that we have both numbers, we can calculate \( B - A \):
- \( B = 3210 \)
- \( A = 1023 \)
- So, \( B - A = 3210 - 1023 \)

### Step 4: Perform the subtraction

1. Let's perform the subtraction:
- \( 3210 - 1023 \)
- Start from the rightmost digit:
- \( 0 - 3 \) (we need to borrow, so we take 1 from the next digit)
- \( 10 - 3 = 7 \)
- Next, \( 0 - 2 \) (again we need to borrow)
- \( 10 - 2 = 8 \)
- Now, we have \( 2 - 0 = 2 \)
- Finally, \( 3 - 1 = 2 \)
- Therefore, \( B - A = 2187 \)

### Final Answer

Thus, the result of \( B - A \) is \( 2187 \).

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