[EM] Truncations Math

[DEMOREP1 at aol.com]

For 5 choices and rank voting there are the following possible combinations --

                Place
              1    2   3    4    5 
Vote 5    5 x 4 x 3 x 2 x 1 = 120*
Vote 4    5 x 4 x 3 x 2       = 120*
Vote 3    5 x 4 x 3             = 60
Vote 2    5 x 4                   = 20
Vote 1    5                        =   5
                      Total            205

Since Vote 5 = Vote 4 only one is included in the Total.

Rearranging the factorial math produces
5 (1 + 4 (1 +3 (1 +2))) = 205  or the general formula --

For N choices there are N x (sum of N-1 combinations) + N possible 
combinations with truncated votes.

N      combinations
2      2         
3      9         = 3 x 2 + 3
4     40        =  4 x 9 + 4
5     205      =  5 x 40 + 5
6     1236     = 6 x 205 + 6
7     8659      = 7 x 1236 + 7

Thus, yet another reason to have computer programs even with 4 choices.   
Since trucated votes can be expected when there are more choices (especially 
with 5 or more choices), the trucation math becomes important.

I have suggested that truncated choices should be deemed to be in a last 
place tie.

Example-
5 Choices, a voter votes for 2, the other 3 are in a last place (fifth 
choice) tie.