Hand patterns in a card game A deck of cards is the Cartesian product of S suits and R ranks.
Many, though not all, card games use S=4 and R∊{6,8,13}.
A hand of H cards is dealt from the deck.
Its distribution, a.k.a. "hand pattern", is an array that describes
how many cards you got from each suit, ignoring suit order (so, it's like a multi-set).
Given a distribution D satisfying len(D)=S, 1≤sum(D)=H≤S×R, 0≤D[i]≤R, D[i]≥D[i+1],
find the probability of it occurring.

Input: an integer R and an array D.

Output: the probability with at least 5 digits after the decimal mark;
 trailing zeroes may be skipped; scientific notation is ok.

Loopholes forbidden. Shortest wins.

Tests:

R    D               probability
13   4 4 3 2     -&gt;  0.2155117564516334148528314355068773
13   5 3 3 2     -&gt;  0.1551684646451760586940386335649517
13   9 3 1 0     -&gt;  0.0001004716813294328274372174524508
13   13 0 0 0    -&gt;  0.0000000000062990780897964308603403
8    3 2 2 1     -&gt;  0.4007096203759162602321667950144035
8    4 2 1 1     -&gt;  0.1431105787056843786543452839337155
8    2 2 1 0     -&gt;  0.3737486095661846496106785317018910
8    3 1 1 0     -&gt;  0.2135706340378197997775305895439377
15   4 4 3 2 1   -&gt;  0.1428926269185580521441708109954798
10   3 0 0       -&gt;  0.0886699507389162561576354679802956
10   2 1 0       -&gt;  0.6650246305418719211822660098522167
10   1 1 1       -&gt;  0.2463054187192118226600985221674877


See also Bridge hand patterns in Wikipedia.

EDIT: dropped unnecessary restriction H≤R

EDIT: added constraint H≥1
APL (Dyalog Unicode), 30 chars



×/!⍨,z,1÷((z←!∘≢⊢)⌸⊢),×∘≢!⍨1⊥⊢


Try it online!

Using @orlp’s formula.