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

Woodall, Douglas R., A note on Fisher's inequality, J. Comb. Theory, Ser. A 77, No.1, 171-176, TA962729 (1997). ZBL0878.05011. [1]: https://pdfs.semanticscholar.org/f8ac/82fa89f026a5ced840f97086f4b40d370bbd.pdf

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

One line proof Fisher's inequality is given by Renaud Palisse

Palisse, Renaud, A short proof of Fisher's inequality, Discrete Math. 111, No.1-3, 421-422 (1993). ZBL0788.04003.

Woodall, Douglas R., A note on Fisher's inequality, J. Comb. Theory, Ser. A 77, No.1, 171-176, TA962729 (1997). ZBL0878.05011.