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2)How many arrangements of the integers 1,2,3,...,n are there such that each integer differs by one (except for the first integer) from some integer to the left of it?




Let's construct the arrangement.
First consider 1, it can only have two placed next to it so it has to be at the end or beginning. Do you see why?
Ok, so let's suppose the arrangement starts with 1, then the only number that can come after it is 2, then 3 then 4...
Now suppose 1 is at the end. Then the only number that can come before is 2, then 3 then 4...
The only two arrangements are
1 2 3 4 5 ... n-1 n and
n (n-1) (n-2) ... 3 2 1
By the way, we are not constructing a digit, but rather an arrangement for the first n integers.

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