Roll 5 12-sided fair die with faces numbered 0 to 11. Expected value of second highest value rolled?

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import random
def add(xx, yy):
    return list(map(lambda a, b: a + b, xx, yy))
x=[0,1,2,3,4,5,6,7,8,9,10,11]
n=100
cc=[0,0,0,0,0]
p=[0,0,0,0,0]
#monte carlo part
for i in range(n):
   for k in range(5):
     p[k]=x[random.randint(0,11)]
   c=sorted(p)
   cc=add(cc,c)
for k in range(5):
  print (cc[k]/float(n))
#analitic part
cc=[0,0,0,0,0]
for x1 in x:
  for x2 in x:
    for x3 in x:
      for x4 in x:
        for x5 in x:
          c=sorted([x1,x2,x3,x4,x5])
          cc=add(cc,c)
print(cc)
for k in range(5):
  print (cc[k]/float(12*12*12*12*12))

I use Python.

$\text{Expectation}[x,x\overset{\sim}{\sim}\text{OrderDistribution}[\{\text{DiscreteUniformDistribution}[\{0,11\}],5\},4]]$

or

$\text{Mean}[\text{Table}[\text{Sort}[t][[4]],\{t,\text{Tuples}[\text{Range}[0,11],5]\}]]$

In either case the reduced fraction is $\frac{77759}{10368}$ .