Marios tosses 6 fair coins and Michael tosses 5 fair coins. What is the probability that Marios tosses more "heads" than Michael?

I notice that :
1/1 + 1/3 = 4/3 4^2 + 3^2 = 5^2
1/3 + 1/5 = 8/15 8^2 + 15^2 = 17^2
1/5 + 1/7 = 12/35 12^2 + 35^2 = 37^2

i need to prove a generalization of the problem.

there are two cases : either Marios tosses more "heads" than Michael or tosses more "tails" . but NOT both. these two cases seem symmetric and each occurs with probability 1/2 . please if its possible help me out. this is an interesting problem that came into my life.

Sincerely M.M.
p.s. Thx for the best of the best pages i ever seen in my life on math subject. keep on the good work!! :)