Consider the 3-lists taken from [3]. How many are there in which each element of [3] appears at least once?
I get .
Answer the same question, but for 4-lists and 5-lists taken from [3].
For the 4-list I get *3.
For the 5-list I get * .
Are these answers correct?
[3] is the set of all positive integers less than or equal to 3.
The 3-lists would then be 123 132 213 231 312 and 321.
A list is an ordered sequence of objects, and a k-list is a list of length k.
Lists are also called words. Can also be written (1,2,3) etc. Without the constraint that the elements can only appear once, (1,1,1) would also be a 3-list of [3].
Yes, that would be true except for the constraint that every element must appear at least once. ( I slightly mistated this constraint in my prior post.)
Since every element must appear at least once, for the 3-list the lists would be (1,2,3) (1,3,2) (2,1,3) (2,3,1) (3,1,2) and (3,2,1); without the constraint we could add (1,1,1) (1,1,2) (1,1,3) etc.
Well then, I think in the case of 4-list from [3] we want to count the number of surjections from a set of four to a set of three. There are 36 of those.
Now with this understanding <3,2,1,1> is different from <1,2,3,1>.
If it means that those are not different then the answer in .