Optimus Template

With just two candidates you could vote 4 different ways: C1,C2,C1C2, and C2C1, and with three candidates you could vote 15 different ways:
C1,C2,C3, C1C2,C1C3,C2C1,C2C3,C3C1,C3C2,C1C2C3,C1C3C2,C2C1C3,C2C3C1,C3C1C2, and C3C2C1.  In mathematical terms, if there are n candidates, the number of different ways ballots could occur is nP1+nP2+nP3+...+nPn, where nPm is the number of permutations of n objects chosen m at a time.  As you can see from the table below, the number of distinct ballot permutations grows very rapidly as the number of candidates increases.  

# Candidates                              # Distinct Ballots

1                   1
2                   4
3                   15
4                   64
5                  325
6                 1,956
7                 13,699
8                109,600
9                986,409
10               9,864,100
11              108,505,111
12             1,302,061,344
13             16,926,797,485
14            236,975,164,804
15           3,554,627,472,075
16           56,874,039,553,216
17          966,858,672,404,689
18         17,403,456,103,284,420
19        330,665,665,962,403,999
20       6,613,313,319,248,080,000