I don't seem to get this question.. what exactly is going on? Do we start with an empty sequence, pick a card from the deck (let's say 2 Heart), put it inside the sequence, so our sequence is now (2 Heart), put it back into the deck, shuffle and pick another card (let's say Queen Spade) so our sequence is now (2 Heart, Queen Spade) and so on n times? and the question is in how many of the possible sequences all the Jacks appear at least once? How would I approach this question?
Apply Principle of Inclusion Exclusion.
Number of all possible sequences $ = 52^n \,$ as each element in the sequence can be one of the $52$. $(n \ge 4)$
Now there are $51^n$ sequences out of these where a specific jack is missing (but it will also have sequences where other jacks are missing).
Similarly $50^n$ sequences where two specific jacks are missing, $49^n$ where three jacks are missing and $48^n$ where all $4$ jacks are missing.
So applying P.I.E to take out overcounting, number of sequences where you have one or more jacks missing
$ = {4 \choose 1} \times 51^n - {4 \choose 2} \times 50^n + {4 \choose 3} 49^n - {4 \choose 4}48^n$
Now if we subtract these from $52^n$, you get all possible sequences with all jacks at least appearing once in the sequence.