This paper presents asymptotic formulas for the abundance of binomial, generalized binomial, multinomial, and generalized multinomial coefficients having any given degree of prime-power divisibility. In the case of binomial coefficients, for a fixed prime
p
p
, we consider the number of
(
x
,
y
)
(x, y)
with
0
≤
x
,
y
>
p
n
0 \leq x, y > p^n
for which
(
x
+
y
x
)
\binom {x+y}{x}
is divisible by
p
z
n
p^{zn}
(but not
p
z
n
+
1
p^{zn+1}
) when
z
n
zn
is an integer and
α
>
z
>
β
\alpha > z > \beta
, say. By means of a classical theorem of Kummer and the probabilistic theory of large deviations, we show that this number is approximately
p
n
D
(
(
α
,
β
)
)
p^{n D((\alpha , \beta ))}
, where
D
(
(
α
,
β
)
)
:=
sup
{
D
(
z
)
:
α
>
z
>
β
}
D((\alpha , \beta )) := \sup \{ D(z) : \alpha > z > \beta \}
and
D
D
is given by an explicit formula. We also develop a “
p
p
-adic multifractal” theory and show how
D
D
may be interpreted as a multifractal spectrum of divisibility dimensions. We then prove that essentially the same results hold for a large class of the generalized binomial coefficients of Knuth and Wilf, including the
q
q
-binomial coefficients of Gauss and the Fibonomial coefficients of Lucas, and finally we extend our results to multinomial coefficients and generalized multinomial coefficients.