Problem:
Score : 400 points

Problem Statement

You are given N items.

The value of the i-th item (1 \leq i \leq N) is v_i.

Your have to select at least A and at most B of these items.

Under this condition, find the maximum possible arithmetic mean of the values of selected items.

Additionally, find the number of ways to select items so that the mean of the values of selected items is maximized.

Constraints

1 \leq N \leq 50

1 \leq A,B \leq N

1 \leq v_i \leq 10^{15}

Each v_i is an integer.

Input

The input is given from Standard Input in the following format:

N A B
v_1
v_2
...
v_N

Output

Print two lines.

The first line should contain the maximum possible arithmetic mean of the values of selected items. The output should be considered correct if the absolute or relative error is at most 10^{-6}.

The second line should contain the number of ways to select items so that the mean of the values of selected items is maximized.

Sample Input 1

5 2 2
1 2 3 4 5

Sample Output 1

4.500000
1

The mean of the values of selected items will be maximized when selecting the fourth and fifth items. Hence, the first line of the output should contain 4.5.

There is no other way to select items so that the mean of the values will be 4.5, and thus the second line of the output should contain 1.

Sample Input 2

4 2 3
10 20 10 10

Sample Output 2

15.000000
3

There can be multiple ways to select items so that the mean of the values will be maximized.

Sample Input 3

5 1 5
1000000000000000 999999999999999 999999999999998 999999999999997 999999999999996

Sample Output 3

1000000000000000.000000
1

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

;; -*- coding: utf-8 -*-
(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 (set-dispatch-macro-character #\# #\> (lambda (s c p) (declare (ignore c p)) (read s nil (values) t))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

;; BEGIN_INSERTED_CONTENTS
;;;
;;; Compute binomial coefficient by direct bignum arithmetic
;;;
;;; This code ist almost the same as that of alexandria.
;;;

(declaim (inline %multiply-range))
(defun %multiply-range (i j)
  (labels ((bisect (j k)
             (declare (type (integer 1 #.most-positive-fixnum) j k)
                      (values integer))
             (if (< (- k j) 8)
                 (multiply-range j k)
                 (let ((middle (+ j (truncate (- k j) 2))))
                   (* (bisect j middle)
                      (bisect (+ middle 1) k)))))
           (multiply-range (j k)
             (declare (type (integer 1 #.most-positive-fixnum) j k))
             (do ((f k (* f m))
                  (m (1- k) (1- m)))
                 ((< m j) f)
               (declare (type (integer 0 (#.most-positive-fixnum)) m)
                        (type unsigned-byte f)))))
    (bisect i j)))

(declaim (inline factorial))
(defun factorial (n)
  (%multiply-range 1 n))

(defun binomial-coefficient (n k)
  (declare #.OPT
           ((integer 0 (#.most-positive-fixnum)) n k))
  (assert (>= n k))
  (if (or (zerop k) (= n k))
      1
      (let ((n-k (- n k)))
        (when (< k n-k)
          (rotatef k n-k))
        (if (= 1 n-k)
            n
            (floor (%multiply-range (+ k 1) n)
	           (%multiply-range 1 n-k))))))

(defun multiset-coefficient (n k)
  (binomial-coefficient (+ n k -1) k))

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(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)

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;; Body

(defun main ()
  (declare #.OPT)
  (let* ((n (read))
         (a (read))
         (b (read))
         (vs (sort (make-array n :element-type 'uint62 :initial-contents (the list (loop repeat n collect (read)))) #'>))
         (max-score (loop for end from a to b maximize (/ (loop for i below end sum (aref vs i) of-type uint62) end)))
         (prevs (make-array n :element-type 'uint32 :initial-element 1)))
    (declare (uint8 n a b))
    (println (float max-score 1d0))
    (loop for i from 1 below n
          do (if (= (aref vs i) (aref vs (- i 1)))
                 (setf (aref prevs i) (+ (aref prevs (- i 1)) 1))
                 1))
    (let ((chunks (copy-seq prevs))
          (i (- n 1)))
      (declare ((simple-array uint32 (*)) chunks))
      (loop (when (<= i 0)
              (return))
            (loop for j from i downto 0
                  until (= 1 (aref chunks j))
                  do (setf (aref chunks j) (aref chunks i))
                  finally (setf (aref chunks j) (aref chunks i))
                          (setf i (- j 1))))
      (println
       (loop for end from a to b
             for score = (/ (loop for i below end sum (aref vs i) of-type uint62) end)
             when (= score max-score)
             sum (binomial-coefficient (aref chunks (- end 1)) (aref prevs (- end 1)))
             of-type uint62)))))

#-swank(main)