We have $5$ objects: A,B,C,D,E.
The number of permutation possible is $5!$.
But I have some constraints:
The first and the last position can only be occupied by A, B or C
There can be a repetition at the end of A, B, C (for example A, B, C, D, E, A)
To resume: If A,B,C are entry or exit points (considering it's a chain), how many possible chains are possible?
EDIT: Thanks lulu, rules indeed are those: To be clear, the rules I am guessing are: either you have a string of length 5 or length 6. The first type are just permutations of your five letters. The second type has a redundant A,B,C at the end. In both cases we require that the first and last character be one of A,B,C
There are two cases. If the string is of length $5$, so a permutation of ABCDE starting and ending with A, B or C, there are $3$ possibilities for the first letter. For each of these there are $2$ possibilities for the last letter. Now for each permutation of first and last letter there are $3!$ ways to arrange the middle three letters, giving $3\times2\times 3!$.
For the strings of length $6$ there are $3$ choices for the first letter and $3$ for the last letter (since these may be the same). Now the remaining letters are the four missing letters if the first and last are the same; otherwise they are the three missing letters plus another copy of the last letter. In either case there are four different letters to arrange in the middle, so $4!$ possibilities.
Overall, there are $3\times2\times 3!+3\times3\times4!$ suitable strings.