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 Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

(eval-when (:compile-toplevel :load-toplevel :execute)
  (defparameter OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0)))
  #+swank (progn (ql:quickload '(:cl-debug-print :fiveam))
                 (shadow :run)
                 (use-package :fiveam)))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

;; BEGIN_INSERTED_CONTENTS
(defmacro nlet (name args &body body)
  (labels ((ensure-list (x) (if (listp x) x (list x))))
    (let ((args (mapcar #'ensure-list args)))
      `(labels ((,name ,(mapcar #'car args) ,@body))
         (,name ,@(mapcar #'cadr args))))))

;; (with-memoizing (:hash-table :test #'equal :key #'cons)
;;   (defun ...))
;; (with-memoizing (:array (10 10 * 10) :initial-element -1 :element-type 'fixnum)
;;   (defun ...))
(defmacro with-memoizing (cache-attribs def-form)
  (let* ((cache-attribs (if (atom cache-attribs) (list cache-attribs) cache-attribs))
         (cache-type (first cache-attribs))
         (dimensions-with-* (when (eql cache-type :array) (second cache-attribs)))
         (dimensions (remove '* dimensions-with-*))
         (rank (length dimensions))
         (rest-attribs (ecase cache-type
                         (:hash-table (cdr cache-attribs))
                         (:array (cddr cache-attribs))))
         (key (prog1 (getf rest-attribs :key) (remf rest-attribs :key)))
         (cache-form (case cache-type
                       (:hash-table `(make-hash-table ,@rest-attribs))
                       (:array `(make-array ',dimensions ,@rest-attribs))))
         (initial-element (when (eql cache-type :array)
                            (assert (member :initial-element rest-attribs))
                            (getf rest-attribs :initial-element))))
    (let ((cache (gensym))
          (value (gensym))
	  (present-p (gensym))
          (name-alias (gensym))
	  (args-lst (gensym))
          (indices (loop repeat rank collect (gensym))))
      (labels ((make-cache-check-form (cache-type args)
                 (case cache-type
                   (:hash-table
                    `(let ((,args-lst (funcall ,(or key #'list) ,@args)))
                       (multiple-value-bind (,value ,present-p)
                           (gethash ,args-lst ,cache)
                         (if ,present-p
                             ,value
                             (setf (gethash ,args-lst ,cache)
                                   (,name-alias ,@args))))))
                   (:array
                    (let ((memoized-args (loop for dimension in dimensions-with-*
                                               for arg in args
                                               unless (eql dimension '*)
                                               collect arg)))
                      (if key
                          `(multiple-value-bind ,indices
                               (funcall ,key ,@memoized-args)
                             (let ((,value (aref ,cache ,@indices)))
                               (if (eql ,initial-element ,value)
                                   (setf (aref ,cache ,@indices)
                                         (,name-alias ,@args))
                                   ,value)))
                          `(let ((,value (aref ,cache ,@memoized-args)))
                             (if (eql ,initial-element ,value)
                                 (setf (aref ,cache ,@memoized-args)
                                       (,name-alias ,@args))
                                 ,value)))))))
               (make-reset-form (cache-type)
                 (case cache-type
                   (:hash-table `(setf ,cache (make-hash-table ,@rest-attribs)))
                   (:array `(prog1 nil
                              (fill (array-storage-vector ,cache) ,initial-element)))))
               (make-reset-name (name)
                 (intern (format nil "RESET-~A" (symbol-name name))))
               (extract-declarations (body)
                 (remove-if-not (lambda (form) (eql 'declare (car form))) body)))
        (ecase (car def-form)
          ((defun)
           (destructuring-bind (_ name args &body body) def-form
             (declare (ignore _))
             `(let ((,cache ,cache-form))
                (defun ,(make-reset-name name) () ,(make-reset-form cache-type))
                (defun ,name ,args
                  (labels ((,name-alias ,args ,@body))
                    (declare (inline ,name-alias))
                    ,@(extract-declarations body)
                    ,(make-cache-check-form cache-type args))))))
          ((nlet)
           (destructuring-bind (_ name bindings &body body) def-form
             (declare (ignore _))
             `(let ((,cache ,cache-form))
                (nlet ,name ,bindings
                      ,(let ((args (mapcar (lambda (x) (if (atom x) x (car x))) bindings)))
                         `(labels ((,name-alias ,args ,@body))
                            (declare (inline ,name-alias))
                            ,@(extract-declarations body)
                            ,(make-cache-check-form cache-type args)))))))
          ((labels flet)
           (destructuring-bind (_ definitions &body labels-body) def-form
             (declare (ignore _))
             (destructuring-bind (name args &body body) (car definitions)
               `(let ((,cache ,cache-form))
                  (,(car def-form)
                   ((,(make-reset-name name) () ,(make-reset-form cache-type))
                    (,name ,args
                           (labels ((,name-alias ,args ,@body))
                             (declare (inline ,name-alias))
                             ,@(extract-declarations body)
                             ,(make-cache-check-form cache-type args)))
                    ,@(cdr definitions))
                   (declare (ignorable #',(make-reset-name name)))
                   ,@labels-body))))))))))

;; (test with-memoizing
;;   (finishes (macroexpand `(with-memoizing (:hash-table :test #'equal)
;;                             (defun add (x y) (+ x y)))))
;;   (finishes (macroexpand `(with-memoizing (:array '(10 10)
;;                                            :element-type 'fixnum
;;                                            :initial-element -1)
;;                             (defun add (x y) (+ x y)))))
;;   (finishes (macroexpand `(with-memoizing (:array '(10 10)
;;                                            :element-type 'fixnum
;;                                            :initial-element -1)
;;                             (labels ((add (x y) (+ x y))
;; 		                     (my-print (x) (print x)))
;; 	                      (add 1 2))))))

(declaim (inline split-ints-into-vector))
(defun split-ints-into-vector (string dest-vector &key (offset 0))
  (declare (string string)
           ((simple-array * (*)) dest-vector)
           ((integer 0 #.most-positive-fixnum) offset))
  (loop for idx from offset below (length dest-vector)
        for pos1 = 0 then (1+ pos2)
        for pos2 = (position #\space string :start pos1 :test #'char=)
        do (setf (aref dest-vector idx)
                 (parse-integer string :start pos1 :end pos2))
        finally (return dest-vector)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(defmacro println (obj &optional (stream '*standard-output*))
  `(let ((*read-default-float-format* 'double-float))
     (prog1 (princ ,obj ,stream) (terpri ,stream))))

;; Hauptteil

(defun main ()
  (declare #.OPT)
  (let* ((n (read))
         (/n (/ (float n 1d0)))
         (as (make-array n :element-type 'uint2)))
    (declare (uint16 n))
    (split-ints-into-vector (read-line) as)
    (println
     (with-memoizing (:array (301 301 301)
                      :element-type 'double-float
                      :initial-element -1d0)
       (nlet dp ((x (count 1 as))
                 (y (count 2 as))
                 (z (count 3 as)))
         (if (and (zerop x) (zerop y) (zerop z))
             0d0
             (/ (* n (+ 1d0
                        (if (zerop x)
                            0d0
                            (* x /n (dp (- x 1) y z)))
                        (if (zerop y)
                            0d0
                            (* y /n (dp (+ x 1) (- y 1) z)))
                        (if (zerop z)
                            0d0
                            (* z /n (dp x (+ y 1) (- z 1))))))
                (+ x y z))))))))

#-swank(main)