Hint: Analyze the question, we have to find out a small 4 digit number with 4 different numbers with 7 at one place. We have a spot that is fixed at one place. Now apply logic of highest and lowest numbers and find the answer.

Complete step by step answer:

Given, Digit 7 is always at one's place.

Since, we have to form a 4 - digit number so there will be digits at its ones, tens, hundreds and thousands place.

$\_ ,\_, \_, \_ $

$\uparrow \uparrow \uparrow \uparrow$

Thousands place Hundreds place Tens place Ones place.

The different digits which can be used to make any number are 0,1,2,3,4,5,6,7,8 and 9.

We already know that at one place there is digit 7 so we are left with digits 0,1,2,3,4,5,6,8,9.

Also, digit 0 cannot be placed at thousands places because we need a 4 - digit number.

For Greatest 4 - digit number,

The digit at thousands places should be greatest of 1,2,3,4,5,6,8,9. So, digit 9 should be there.

Also, no digit can repeat. So, now we are left with digits 0,1,2,3,4,5,6,8.The digit at hundreds place should be the greatest of 0,1,2,3,4,5,6,8. Hence, digit 8 will be placed at hundreds places and now, we are left with 0,1,2,3,4,5,6.

The digit at tens place should be greatest of 0,1,2,3,4,5,6.

Therefore, digit 6 will be placed at tens place.

Hence, the greatest four digit number is 9867.

For Smallest 4 - digit number,

The digit at thousands places should be the smallest of 1,2,3,4,5,6,8,9. So, digit 1 should be there.

Here, digit 0 is not used at thousands places as it will convert the number into a three digit number.

Also, no digit can repeat. So, now we are left with digits 0,2,3,4,5,6,8,9. The digit at hundreds place should be the smallest of 0,2,3,4,5,6,8,9. Hence, digit 0 will be placed at hundreds places and now, we are left with 2,3,4,5,6,8,9. The digit at tens place should be the smallest of 2,3,4,5,6,8,9.

Therefore digit 2 will be placed at tens place.

Hence, the smallest four digit number is 1027.

Note - Understanding the conditions mentioned in the problem statement is important to avoid error.

One should eliminate all unnecessary options to solve the problem better.