The 120 of your permutations start with 1, the second bunch of 120 starts with 2. Therefore, the 317^th permutation starts with 3.

At this point you may forget about 3 and think of a 5-digits number, of which there are 120. You are interested in position 317 - 240 = 77.

Of these 120, the first 24 start with 1, the second 24 start with 2, the third bunch starts with 4 (3 was out on the first step), the fourth one with 5. Therefore, the second digit is 5.

You are down to 4 digits. And so on.

First Question: In the 720 permutations of {1,2,3,4,5,6},
listed in increasing order, what is the 317th number?
Answer: 351624

Second Question: How did you do that???
Answer: There is a procedure found in 1963 by Dale Kozniuk
(a high school student). It is quite simple, but perhaps
too long for a post here.

I'll send it directly to AlexB. If you are interested,
I'm sure he will forward it to you.