How many positive triples $(n,m,k)$ satisfy the following equation: $$n!+m!=2^k.$$
My try follows, I used trial and examined many values; only $4$ triples work which are $$(1, 1,1), (2,2,2), (2,3,3), (3,2,3).$$
Are there other triples? If not, how can I prove that?
Thank you for your help.