Clear[n, h, p, r];

ans = 
      Sum[n!/(r! (n - r)!)*p^r*(1 - p)^(n - r), {
          r, h, n}]

\(\(\({n, h, p} = {12, 6, 3/6}; 
    \)
 (*\ n : \ dice\ rolled\ *) 
 (*\ h : \ damage\ need\ to\ kill\ *) 
      
 (*\ p : \ probability\ each\ die\ hits\ *) \)\)

ans

N[ans]