Amit has forgotten his 4-digit locker key. He remembers that all the digits are positive integers and are different from each other. Moreover, the fourth digit is the smallest and the maximum value of the first digit is 3. Also, he recalls that if he divides the second digit by the third digit, he gets the first digit.

How many different combinations does Amit have to try for unlocking the locker?

Let the 4-digit locker key be $$abcd$$, where $$9\ge\ a,b,c,d>0$$, and they are all unique.

It is given that $$a\le\ 3\ \&\ \frac{b}{c}=a$$

It is also given that 'd' is the smallest number.

Case (i): $$a=1$$

Not possible, as 'd' is the smallest number, 'a' cannot be 1.

Case (ii): $$a=2$$

Then 'd' can take only one value, i.e. $$d=1$$ 

$$b=2c$$

 $$if\ c=3,\ then\ b=6$$

$$if\ c=4,\ then\ b=8$$

for $$c\ge\ 5$$, 'b' will not be a single-digit number.

Hence, two cases are possible, i.e. (2631, 2841).

Case (iii): $$a=3$$

'd' can take two values, i.e. $$d=1\ or\ d=2\ $$

& $$b=3c$$

If d = 1, then 'c' can take only one value. i.e. $$\ c=2,\ then\ b=6$$

If d = 2, then 'c' cannot take any value.

Hence one case is possible, i.e. (3621)

Amit has to try 3 different combinations. Option (E) is correct.