Arrangement of Tokens in 2 identical Boxes

 There are n number of tokens marked 1,2,3,4,..........n respectively. One has to pick up 2 tokens and put them in 2 boxes, one in each. None of the boxes should  remain empty nor have more than 1 token. Both the boxes are identical. In how many ways, 2 tokens can be allocated to both boxes out of n available tokens ? Ans : C(n,2) What is the minimum sum of 2 tokens so allocated ? Ans - 1+2=3 What is the maximum sum of 2 tokens so allocated ? Ans:- n+(n-1) = 2n-1 What is the average sum of 2 tokens so allocated ? xav=[3+(2n-1)] /2 = n+1 In how many ways, the 2 tokens can be arranged so as to get the average sum? Ans:-quotient (n/2) If X is the sum of 2 tokens such that X=xav ±x, then no. of arrangements to get X is same for  xav+x & xav-x. If n is odd, no. of arrangements to get a number equal to  xav ±1=No. of arrangements to get xav If n is even, no. of arrangements to get a number equal to  xav ±1=(No. of arrangements to get xav) -1. Example-- No. of tokens 1,2,3,.......8. One token in each of 2 boxes. Minimum no. that can be put--3, maximum no. that can be put--15, Average no. --09 Token with   sum-3(xav-6)           sum-4(xav-5)      sum-5(xav-4)       sum-6(xav-3)        sum-7(xav-2)        sum-8(xav-1)          sum-9 (xav=9+x=0) 1+2                          1+3                       1+4                     1+5                      1+6                      1+7                         1+8                                                                                          2+3                    2+4                      2+5                       2+6                         2+7                                                                                                                                                 3+4                       3+5                         3+6                                                                                                                                                                                                             4+5 sum-10(xav+1)     sum-11(xav+2)    sum-12 (xav+3)    sum-13(xav+4)      sum-14(xav+5)     sum-15(xav+6) 2+8                       3+8                        4+8                       5+8                         6+8                     7+8                        3+7                       4+7                        5+7                      6+7                         4+6                      5+6

 No. of tokens Sum of tokens in 2 boxes is no. of boxes no. of ways total no. of  tokens can be allocated in 2 boxes Minimum value of sum Maximum value of sum Average value of sum no. of ways of allocation of 2 tokens to get average sum no. of different arrangements to get the sum as above