For a 4-digit number (greater than 1000), sum of the digits in the thousands, hundreds, and tens places is 15. Sum of the digits in the hundreds, tens, and units places is 16. Also, the digit in the tens place is 6 more than the digit in the units place. The difference between the largest and smallest possible value of the number is

The correct option is 811.

Let the 4-digit number be represented as:
$$N=\overline{abcd}$$
where:
• $a$ is the digit in the thousands place (where $1\le a\le 9$ since the number is greater than 1000),
• $b$ is the digit in the hundreds place ($0\le b\le 9$),
• $c$ is the digit in the tens place ($0\le c\le 9$), and
• $d$ is the digit in the units place ($0\le d\le 9$).

Based on the problem statement, we can write the following system of equations:
1) The sum of the digits in the thousands, hundreds, and tens places is 15:
$$a+b+c=15$$
2) The sum of the digits in the hundreds, tens, and units places is 16:
$$b+c+d=16$$
3) The digit in the tens place is 6 more than the digit in the units place:
$$c=d+6$$

First, let's subtract the first equation from the second equation:
$$(b+c+d)-(a+b+c)=16-15$$
Simplifying this gives:
$$d-a=1\Rightarrow d=a+1$$

Next, we substitute $d=a+1$ into the third equation:
$$c=(a+1)+6\Rightarrow c=a+7$$

Since $c$ is a single digit, its value cannot exceed 9. Therefore:
$$a+7\le 9\Rightarrow a\le 2$$

Since the number is a 4-digit number greater than 1000, the thousands digit $a$ must be at least 1. Thus, the only possible values for $a$ are 1 and 2.

Let's analyze the two possible cases to find the corresponding digits and numbers:

Case 1: When $a=1$
• $d=a+1=1+1=2$
• $c=a+7=1+7=8$
• To find $b$, substitute $a$ and $c$ into the first equation:
$$1+b+8=15\Rightarrow b=6$$
So, the first possible number is 1682.

Case 2: When $a=2$
• $d=a+1=2+1=3$
• $c=a+7=2+7=9$
• To find $b$, substitute $a$ and $c$ into the first equation:
$$2+b+9=15\Rightarrow b=4$$
So, the second possible number is 2493.

From these two cases, we get:
• The largest possible value of the number is 2493.
• The smallest possible value of the number is 1682.

Now, calculate the difference between the largest and smallest possible values:
$$\text{Difference}=2493-1682=811$$