Happy Monday! We have two great combinatorics puzzles for you this week.
Solutions in pdf bottom of post
Question 1: Exactly 6 [Medium]
If I roll a die 36 times, what is the probability that each number appears exactly 6 times?
Question 2: Path Probability [Medium]
You are randomly traveling through a 3-dimensional grid-space. You started at the point (0,0,0), and reached the point $(r,k,m)$. At each step, you randomly picked one of the three dimensions and moved one space forward. What is the probability that you reached the point (r,k,m) via the following path:
the first r steps take you to (r,0,0)
the next k steps take you to (r,k,0)
the last m steps take you to (r,k,m)
I'm not sure I understand your solution to q2. The way I (think i) solved it is to say:
At each step there is a 1/3 probability of taking the correct step.
The total number of steps to take is n = r+k+m.
Therefore the solution is 1/3^n.
Am I making a stupid mistake somewhere or is this also valid?