  Many combinatorial problems involve determining whether a universe of $n$
elements contains a witness consisting of $k$ elements which have some
specified property. In this paper we investigate the relationship between the
decision and enumeration versions of such problems: efficient methods are known
for transforming a decision algorithm into a search procedure that finds a
single witness, but even finding a second witness is not so straightforward in
general. We show that, if the decision version of the problem can be solved in
time $f(k) \cdot poly(n)$, there is a randomised algorithm which enumerates all
witnesses in time $e^{k + o(k)} \cdot f(k) \cdot poly(n) \cdot N$, where $N$ is
the total number of witnesses. If the decision version of the problem is solved
by a randomised algorithm which may return false negatives, then the same
method allows us to output a list of witnesses in which any given witness will
be included with high probability. The enumeration algorithm also gives rise to
an efficient algorithm to count the total number of witnesses when this number
is small.
