A combinatorial proof of a more general inequality is [given][1] by Douglas Woodall.

<cite authors="Woodall, Douglas R.">_Woodall, Douglas R._, [**A note on Fisher's inequality**](http://dx.doi.org/10.1006/jcta.1996.2729), J. Comb. Theory, Ser. A 77, No.1, 171-176, TA962729 (1997). [ZBL0878.05011](https://zbmath.org/?q=an:0878.05011).</cite>
  [1]: https://pdfs.semanticscholar.org/f8ac/82fa89f026a5ced840f97086f4b40d370bbd.pdf