I want to design a ballot for four elections. Actually all the candidate races are on 1 ballot. I need to know how many different ballot styles would be needed for all of the candidates to be in each rotation an equal number of times.

For example:

  • A,B,C & D are running for mayor

  • E,F,G & H are running for congress

  • I,J & K are running for senate

  • L,M & N are running for governor

They are all on the same ballot. But in each race their name (for instance A) has to be in the #1 rotation, #2 rotation, #3 rotation, and #4 rotation for his race on this ballot an equal number of times as B,C and D.

The same goes for the other candidates for their perspective races.

How many total ballot styles will there be?


Hi Kelley,

If there were only one election for mayor then there are 24 ballot styles. The reason is as follows. For the first position on the ballot you have 4 choices, A, B, C or D. After you make a choice for the first position you have 3 choices remaining for the second position. Then you have 2 choices for the third position and only one remaining choice for the fourth position. Thus in total you have 4x3x2x1 = 24 different ballots. We write this number 4! and call it "4 factorial". That is 4! = 4x3x2x1.

At this point you can proceed in two different ways. One way is to introduce two new candidates, blank1 in the senatorial race and blank2 in the race for governor. Consider the 24 ballot styles you have produced for the race for mayor. For each one, produce a ballot style for the 4 races by duplicating the arrangement of A, B, C and D in each of the remaining 3 races and then replacing

  • E for A in the race for congress, I for A in the race for senate and L for A in the race for governor

  • F for B in the race for congress, J for B in the race for senate and M for B in the race for governor

  • G for C in the race for congress, K for C in the race for senate and N for C in the race for governor

  • H for D in the race for congress, blank1 for D in the race for senate and blank2 for D in the race for governor

When you print the ballots just delete blank1 and blank2. This gives you 24 ballot styles and in each race, each candidate's name appears in each position the same number of times as each other candidate in the same race.

These ballots do have some regularity in that every ballot that has A first on the mayor list also has E first on the congress list, and so on. If you want the lists to operate independently then you need many more ballot styles. In this case you would need 4!x4!x3!x3! = 20736 styles.


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