The computational complexity of counting the number of matchings of size k in a given triple set remains open, and it is conjectured that the problem is infeasible. In this paper, we present a fixed parameter tractable randomized approximation scheme (FPTRAS) for the problem. More precisely, we develop a randomized algorithm that, on given positive real numbers e and δ, and a given set S of n triples and an integer k, produces a number h in time O(5.483kn2 ln(2/δ)/e2) such that prob[(1 - e)h0 ≤ h ≤ (1 + e)h0] > 1 - δ where h0 is the total number of matchings of size k in the triple set S. Our algorithm is based on the recent improved color-coding techniques and the Monte-Carlo self-adjusting coverage algorithm developed by Karp and Luby.