I have to find out the possible number combinations which sum up to $X$ using $Y$ specefic digits only, as in $Y$ means using $3$ digits, that too specefied such $1,2,3$ only. For example, in a specefic case where, $X = 5$ and $Y = \{1,2,3\}$ only so my possible number of combinations, keeping in mind, every different combination will also be counted different would be as follows:
(1 1 1 1 1)
(1 1 1 2)
(1 1 2 1)
(1 2 1 1)
(2 1 1 1)
(1 2 2)
(2 2 1)
(2 1 2)
(1 1 3)
(1 3 1)
(3 1 1)
(2 3)
(3 2)
which now sums up to be $13$, I came up with something although it was related to fibonacci series, any specefic relation with the above.
HINT:
First look at the different ways you can partition $X$ (Ferrer diagrams can be helpful to pick specific ones "having only some specific integers")
Then you can use permutation of a multiset to find the number of arrangements.
Hope this helped