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. |