open System
open System.Collections.Generic 
 
let isPrime n =
  List.exists (fun i -> n % i = 0) [2..(n - 1)]
 
let recip =
  let cache = Dictionary<_, _>()
  fun n m ->
      assert (isPrime m)
      if cache.ContainsKey( (n, m) ) then cache.[ (n, m) ]
      else 
        let res = List.find (fun a -> n * a % m = 1) [1..(m - 1)]
        cache.[ (n, m) ] <- res
        res
 
let rec binomialMod n k m  =
  let rec recBinomialMod n k m result =
    if k = 1 then n * result % m 
    else 
      let k_1 = recip k m
      recBinomialMod (n - 1) (k - 1) m (result * n * k_1 % m)      
  recBinomialMod n k m 1
 
let primes = List.filter isPrime [1000..5000]
let binomials = List.map (fun m -> binomialMod (int (1000000000000L % (int64 m))) 1000000000 m) primes
 
let solution =
  let rec recSolution (binomials:(int list)) product chosen =
    seq {      
      if chosen = 3 then yield product
      else if binomials.IsEmpty then yield 0 
      else
        yield! recSolution binomials.Tail product chosen
        yield! recSolution binomials.Tail (product * binomials.Head) (chosen + 1)
    }
  (Seq.sum (recSolution binomials 1 0))
 
Console.WriteLine solution

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MQoyCjEwCjQyCjExCg==

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