p[a_, b_, c_, k_, 0] := 0;
p[a_, b_, c_, k_, t_] := p[a, b, c, t, k] =
   ((Binomial[a + b + c, k] - Binomial[b + c, k]) p[a - 1, b, c, k, t - 1] +
    (Binomial[b + c, k] - Binomial[c, k]) 1 +
     Binomial[c, k] p[a, b, c - 1, k, t - 1])/Binomial[a + b + c, k];

p[2,1,97,10,20]

7641626974773994684341551557/8631114035410540271032419432