If I have 6 regular dice, (each numbered 1-6):
What is the probability that when rolled that each will be a different number.(each individual di is a different number from 1-6, but a random order)
What is the probability that of the six rolls, each will be an increasing number starting with 1. (First roll is 1, second is 2, etc.)
I came up with this problem after watching an episode of numberphile. Thanks!
On
For a hand-waving solution to Part 1:
Roll the first die; any number is acceptable; probability is $\frac66$ or $1$
Roll the second die; only five acceptable numbers left; probability is $\frac56$
Roll the third die; only four acceptable numbers left: probability is $\frac46$
More of the same for the last three dice; multiply each probability together, giving$$P_{different}=\frac{6\times 5\times 4\times 3\times 2\times 1}{6\times 6\times 6\times 6\times 6\times 6}$$which of course matches the correct answer of @Dr Xorile
There are $6^6$ ways of throwing the dice in total. There are $6!$ ways of throwing all the numbers from 1 to 6 in any order. There is only one way of throwing $1,2,3,4,5,6$.
Hence your first answer is $6!/6^6$, and your second is $1/6^6$.