Counting Ballots? How would I count the ballots in this scenario? A ballot lists ten candidates for city council, eight candidates for the school board, and five bond issues. The ballot instructs voters to choose up to four people running for city council, rank up to three candidates for the school board, and approve or reject each bond issue. How many different ballots can be cast if partially completed (or empty) ballots are allowed? I was thinking 10 choose 4 * 8 choose 3 * 5 choose 5, but I'm not sure if 5 choose 5 would work for the bond issues and the way I have it set up would account for partially completed or empty ballots.

Unfortunately all three components you calculated are incorrect.

1. ${10 \choose 4}$ insists on exactly four votes, not up to 4. You want ${10\choose 4}+{10\choose 3}+{10\choose 2}+{10\choose 1}+{10\choose 0}$.

2. ${8\choose 3}$ does not rank the three candidates, only chooses them. What you want is $(8)_3+(8)_2+(8)_1+(8)_0=8\times 7\times 6+8\times 7+8+1$.

3. ${5\choose 5}$ does not do anything, equalling 1. Ballots may be blank, hence each bond issue has three choices (accept, reject, blank). Hence, what you want is $3^5$.