Problem:
Score : 100 points

Problem Statement

There are N dishes, numbered 1, 2, \ldots, N.
Initially, for each i (1 \leq i \leq N), Dish i has a_i (1 \leq a_i \leq 3) pieces of sushi on it.

Taro will perform the following operation repeatedly until all the pieces of sushi are eaten:

Roll a die that shows the numbers 1, 2, \ldots, N with equal probabilities, and let i be the outcome. If there are some pieces of sushi on Dish i, eat one of them; if there is none, do nothing.

Find the expected number of times the operation is performed before all the pieces of sushi are eaten.

Constraints

All values in input are integers.

1 \leq N \leq 300

1 \leq a_i \leq 3

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 \ldots a_N

Output

Print the expected number of times the operation is performed before all the pieces of sushi are eaten.
The output is considered correct when the relative difference is not greater than 10^{-9}.

Sample Input 1

3
1 1 1

Sample Output 1

5.5

The expected number of operations before the first piece of sushi is eaten, is 1.
After that, the expected number of operations before the second sushi is eaten, is 1.5.
After that, the expected number of operations before the third sushi is eaten, is 3.
Thus, the expected total number of operations is 1 + 1.5 + 3 = 5.5.

Sample Input 2

1
3

Sample Output 2

3

Outputs such as 3.00, 3.000000003 and 2.999999997 will also be accepted.

Sample Input 3

2
1 2

Sample Output 3

4.5

Sample Input 4

10
1 3 2 3 3 2 3 2 1 3

Sample Output 4

54.48064457488221

Solution in LuaJIT (2.0.4) (judge verdict: Accepted)

local n = io.read("*n")
local a1, a2, a3 = 0, 0, 0
for i = 1, n do
  local a = io.read("*n")
  if a == 1 then a1 = a1 + 1
  elseif a == 2 then a2 = a2 + 1
  else a3 = a3 + 1
  end
end
local t = {}
local p = {}
for i = 1, n + 1 do
  local r0 = i - 1
  t[i] = {}
  p[i] = {}
  for j = 1, n + 2 - i do
    local r1 = j - 1
    t[i][j] = {}
    p[i][j] = {}
    for k = 1, n + 3 - i - j do
      local r2 = k - 1
      local r3 = n - r0 - r1 - r2
      t[i][j][k] = 0
      p[i][j][k] = 0
      if r0 == 0 and r1 == a1 and r2 == a2 then
        p[i][j][k] = 1
      elseif 3 < i + j + k then
        local p1, t1 = 0, 0
        if 1 < i then
          p1 = p[i - 1][j + 1][k] * (r1 + 1) / (r1 + 1 + r2 + r3)
          t1 = t[i - 1][j + 1][k] + n / (n - (r0 - 1))
        end
        local p2, t2 = 0, 0
        if 1 < j then
          p2 = p[i][j - 1][k + 1] * (r2 + 1) / (r1 + r2 + r3)
          t2 = t[i][j - 1][k + 1] + n / (n - r0)
        end
        local p3, t3 = 0, 0
        if 1 < k then
          p3 = p[i][j][k - 1] * (r3 + 1) / (r1 + r2 + r3)
          t3 = t[i][j][k - 1] + n / (n - r0)
        end
        p[i][j][k] = p1 + p2 + p3
        if 0 < p[i][j][k] then
          t[i][j][k] = (p1*t1 + p2*t2 + p3*t3) / (p1 + p2 + p3)
        end
      end
    end
  end
end

print(t[n + 1][1][1])