Correct option is D
Solution:
Let's proceed by understanding what could be the possible outcomes after N steps.
If one-step is taken, the person could end up anywhere (East, West, South, North) so d = L.
If two-steps are taken, the person could move two steps back to the origin (move East then West, or move North then South), move a step East or West and then a step North or South, or move two steps in the same direction.
Therefore, the possible distances are 0L, √2L, or 2L.
If three-steps are taken, the person could return to the origin (e.g., East, West, North), or make several other moves producing distances of √2L, √3L, √5L.
If four-steps are taken, there are more possibilities. In this problem, we are only interested in the possibilities where the total displacement vector's magnitude d ≤ 3L.
Considering all possible combinations of moves in four steps, we get four situations: Four moves in the same direction (e.g., North, North, North, North).
This leads to a distance of 4L.
Three moves in the same direction and one move in a different plane (e.g., North, North, North, East).
This can lead to distances of √10L, √2L.
Two pairs move in the same direction but in different planes (e.g., North, North, East, East). This leads to a distance of 2√2L.
Two moves in one direction and the other two moves in two different directions (e.g., North, North, East, South). This can give √5L, 1L, √2L.
Four moves in four different directions (e.g., North, South, East, West). This leads to a distance of 0L.
From the above, we can notice that distances greater than 3L (i.e., √5L, √10L and 4L) can only occur in situations 1 and 2.
So, let's calculate the probabilities.
The total number of outcomes in this situation is 4 (one for each direction). There are 4×4 ways of choosing which direction to go three times, and 3 ways of choosing the direction for the fourth step. This gives a total of 48 outcomes.
But only 16 of these result in a distance of √2L (8 of the form NNNE and 8 of the form NNNW), while the remaining 32 results in a distance greater than 3L.
This situation always results in a distance less than or equal to 3L. This situation always results in a distance less than or equal to 3L. This situation always results in a distance less than or equal to 3L.
Finally, the total number of outcomes is 44 = 256, and the number of undesired outcomes (where d > 3L) is 4 + 32 = 36.
This gives us a probability
