As it is known that any contestant can enter and exit from any gate, the first contestant can enter from any of the six gates. He latches that door from inside and after completing his task he moves out from another door latching it from outside.

Hence at this position 2 doors are latched, one from inside and the other from outside. When the 2

^{nd} contestant starts his task he enters from a door which was not used by the 1

^{st} contestant and latches it from inside, completes his task and moves from some other door and then latches it from outside.

The 3

^{rd} contestant does the same.

Now at this stage all the six doors are latched, three from inside and three from outside. Hence the next contestant can enter from any of the three doors latched from outside.

As the 4

^{th} contestant enters by opening a door latched from outside and leaves by opening a door latched from inside, at this stage there are 4 latched doors: two latched from outside and other two latched from inside

Now the 5

^{th} contestant enters who latches the doors he uses i.e. if he comes across a latched door and he can open it he uses it and latches it from other side.

So when he starts his task there are two open doors, two doors latched from inside and two doors latched from outside so there arise following cases:

He enters from a door latched from outside and leaves from a door latched from inside

He enters from a door latched from outside and leaves from a door which is not latched

He enters from a door which is not latched and leaves from a door which is not latched

He enters from a door which is not latched and leaves from a door latched from inside

Depending on the above cases the situation after contestant 5 has moved out can be tabulated as follows

Now when the 6

^{th} contestant starts he can select any door at random. His probability of entering the room depends on the doors used by the 5

^{th }contestant.

If 5

^{th} contestant follows case 1 then,

If 5

^{th} contestant follows case 2 then,

If 5

^{th} contestant follows case 3 then,

If 5

^{th} contestant follows case 4 then,

So the probability that the 6

^{th} contestant is able to enter through a randomly selected door is given as follows

Hence, option 3.