Arrangement of Tokens in 3 identical Boxes

There are n number of tokens marked 1,2,3,4,..........n respectively.
One has to pick up 3 tokens and put them in 3 boxes, one in each.
None of the boxes should  remain empty nor have more than 1 token.
All boxes are identical.
In how many ways, 3 tokens can be allocated to 3 boxes out of n available tokens ? Ans : C(n,3)
What is the minimum sum of 3 tokens so allocated ? Ans - 1+2+3=6
What is the maximum sum of 3 tokens so allocated ? Ans:- n+(n-1)+(n-2) = 3n-3=3(n-1)
What is the average sum of 3 tokens so allocated ? xav=[6+(3n-3)] /2 = 3(n+1) /2 . since xav has to be a whole number, for all n=odd no., xav=whole no. and for all n=even number, xav is xav1=floor(xav); or xav2=ceiling(xav); However no. of arrangements in both xav1 and xav2 remain the same.
In how many ways, the 3 tokens can be arranged so as to get the average sum? Ans:- see the worksheet below
If X is the sum of 3 tokens such that X=xav ±x where x is an integer, then no. of arrangements to get X is same for  xav+x & xav-x.
If n is even, no. of arrangements to get a number equal to  xav1 = no. of arrangements to get xav2 and no. of arrangements to get any X=xav1-x =xav2+x
If n is odd, no. of arrangements to get a number equal to  xav +1=No. of arrangements to get xav -1 i.e. X=xav ±x
Example 1-- No. of tokens 1,2,3,.......8. One token in each of 3 boxes. Minimum no. that can be put--6, maximum no. that can be put--21, Average no. --13.5 ->13,14 . Total no. of arrangements is C(8,3) =56. Maximum no. of arrangement is at sum of 13,14 which are average values. Arrangement is 6 in no. At all other sums, the arrangement has lesser value.
Token with sum-6(xav1-7)   sum-7(xav1-6)      sum-8(xav1-5)       sum-9(xav1-4)        sum-10(xav1-3)        sum-11(xav1-2)          sum-12 (xav1-1)
                         1+2+3                   1+2+4                1+3+4                   2+3+4                    2+3+5                     2+4+5                       3+4+5

                           -----                      -----                  1+2+5                   1+3+5                    1+4+5                     2+3+6                       1+5+6

                           ----                       ----                     ----                      1+2+6                    1+3+6                     1+4+6                       2+4+6

                           ----                       ----                     ----                        ----                       1+2+7                     1+3+7                       2+3+7

                           ----                       ----                     ----                        ----                       ----                          1+2+8                       1+4+7

                           ----                      ----                      ----                        ----                        ----                          ----                          1+3+8

                sum-13(xav1)     sum-14(xav2)            sum-15 (xav2+1)    sum-16(xav2+2)      sum-17(xav2+3)     sum-18(xav2+4)  sum-19(xav2+5)  sum-20(xav2+6)  sum-21(xav2+7)
                        3+4+6                   3+5+6                       4+5+6                   4+5+7                    4+6+7                    5+6+7                 4+7+8                5+7+8                6+7+8

                        2+5+6                   3+4+7                       2+6+7                   3+6+7                    2+7+8                    3+7+8                 5+6+8                 ----                     ----

                        2+4+7                   2+5+7                       3+5+7                   1+7+8                    3+6+8                    4+6+8                   ----                    ----                     ----

                        1+5+7                   1+6+7                       1+6+8                   2+6+8                    4+5+8                     ----                       ----                    ----                     ----

                        1+4+8                   1+5+8                       2+5+8                   3+5+8                     ----                         ----                       ----                    ----                     ----

                        2+3+8                   2+4+8                       3+4+8                     ----                         ----                          ----                     ----                    ----                     ---- 

 
 Example 2-- No. of tokens 1,2,3,.......7. One token in each of 3 boxes. Minimum no. that can be put--6, maximum no. that can be put--18, Average no. --13.5 ->12  .Total no. of arrangements - C(7,3) =35                    
Token with sum-6(xav-6)   sum-7(xav-5)      sum-8(xav-4)       sum-9(xav-3)        sum-10(xav-2)        sum-11(xav-1)          sum-12 (xav)
                         1+2+3                   1+2+4                1+3+4              2+3+4                    2+3+5                     2+4+5                       3+4+5

                           -----                      -----                  1+2+5              1+3+5                    1+4+5                     2+3+6                       1+5+6

                           ----                       ----                     ----                  1+2+6                    1+3+6                     1+4+6                       2+4+6

                           ----                       ----                     ----                        ----                   1+2+7                     1+3+7                       2+3+7

                           ----                       ----                     ----                        ----                       ----                        ----                         1+4+7

                           ----                      ----                      ----                        ----                        ----                       ----                         

 

                   sum-13(xav+1)     sum-14(xav+2)      sum-15 (xav+3)      sum-16(xav+4)        sum-17(xav+5)        sum-18(xav+6
                         3+4+6                  3+5+6                       4+5+6                   4+5+7                    4+6+7                    5+6+7                

                        2+5+6                   3+4+7                       2+6+7                   3+6+7                     -----                        ----                      

                        2+4+7                   2+5+7                       3+5+7                   ----                          -----                       -----                    

                        1+5+7                   1+6+7                       -----                      ----                          ----                         ----

                        ----                         ----                           -----                     ----                          -----                         -----

 

The curve of no. of arrangements vrs. sum of tokens is a bell shaped curve with peak at the average sum and symmetrical on both sides of the peak.
 

 

No. of tokens(n)
Sum of tokens in 3 boxes is
no. of boxes
 
no. of ways total no. of  tokens can be allocated in 3 boxes
Minimum value of sum
Maximum value of sum
Range(summax-summin)
Average value of sum (xav1,xav2 if n is even) or else xav -
no. of ways of allocation of 3 tokens to get average sum(A)
Range/A
no. of different arrangements to get the sum as above
   
Deference:1,2,3,4,5,6  
   

 

Work Sheet
no. of tokens

starting from 1

t

    add Maximum no. of arrangements

( which happens when sum of 3 tokens=average value)

arr

+ =     (arr2=arr1+add2)
+ =     (arr3=arr2+add3=arr1+add2+add3)
+ =     (arr4=arr3+add4=arr1+add2+add3+add4)
+ =    
          We find arrX= k=XΣk=1 add k if we take arr1=add1
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