How Many Ways To Empty The Glove Box? Inspired by this glove-themed 538 Riddler Express Puzzle.
Task
You are given a positive integer n, and a list A = [a_1, a_2, ..., a_k] of k distinct positive integers.
Then a restricted composition is an ordered list P = [p_1, p_2, ..., p_m] where each p_i is a (not necessarily distinct) member of A, and p_1 + p_2 + ... + p_m = n.
So, if n = 10, and A = [2,3,4] then an example of a restricted composition would be P = [3,4,3]. Another example would be P = [2,3,3,2]. A third example would be P = [3,3,4]. But there's no restricted composition that starts [3,3,3,...], because 10-(3+3+3) = 1, which is not in A.
We want the total number of different restricted compositions given the inputs, as an integer.
Inputs
A positive integer n and a list A of distinct positive integers. All reasonable input formats allowed.
Output
The number of distinct restricted compositions.
Terms and Conditions
This is code-golf; and thus we seek the shortest submissions in bytes satisfying the constraints. Any use of the usual loopholes voids this contract.
Test Cases
(5, [2, 3, 4]) =&gt; 2
(10, [2, 3, 4]) =&gt; 17
(15, [3, 5, 7]) =&gt; 8
Python 2, 50 49 bytes

-1 byte thanks to @Jonathan Allan



f=lambda n,A:n&gt;=0and(n&lt;1)+sum(f(n-x,A)for x in A)


Try it online!