Computational complexity of counting induced subgraphs which admit perfect matchings
The problem is #P-complete. It follows from the last paragraph of page 2 of the following paper:

C. J. Colbourn, J. S. Provan, and D. Vertigan, The complexity of computing the Tutte
polynomial on transversal matroids, Combinatorica 15 (1995), no. 1, 1–10.

http://www.springerlink.com/content/wk55t6873054232q/