The members of City Council are to vote yes or no on each of seven issues. In marking a ballot, each councilor has the option of abstaining on as many as six of the issues, but should not abstain in all seven cases. In how many ways can the ballot be marked?
3^7 - 1 = 2186
There are 3^7 possible choices of Y,N, and A in seven ballots, but one of them (seven A's) is disallowed.
what is the A for?
I do not get the 3^7 wouldn't it be 7^3 because you have seven questions but can can choose to do one of three for each question?
To find the number of ways the ballot can be marked, we need to consider two scenarios: when at least one councilor abstains from voting on exactly six issues, and when no councilor abstains from voting on six issues.
Scenario 1: At least one councilor abstains from voting on exactly six issues.
In this scenario, we have to choose one councilor to abstain from voting on six out of the seven issues. Since there are multiple councilors who can make this choice, we need to multiply the number of ways each councilor can abstain.
First, let's calculate the number of ways a single councilor can abstain from voting on exactly six issues. Since there are 7 issues, the councilor can choose the 6 abstained issues in 7C6 = 7 ways.
However, we have multiple councilors, and each of them can independently choose to abstain from voting on six issues. Therefore, we need to multiply the individual choices for each councilor. If there are n councilors, we multiply this by n.
So, in scenario 1, the number of ways the ballot can be marked is: 7C6 * n
Scenario 2: No councilor abstains from voting on exactly six issues.
In this scenario, no councilor can abstain from voting on exactly six issues. Therefore, each councilor can either vote 'yes' or 'no' on each of the seven issues, but they should not abstain in all seven cases.
For each of the seven issues, there are two choices - 'yes' or 'no'. Since each councilor can independently choose for each issue, we need to multiply the individual choices for each councilor by 2. If there are n councilors, we multiply this by 2^n.
So, in scenario 2, the number of ways the ballot can be marked is: 2^n
Total number of ways the ballot can be marked is the sum of the two scenarios: 7C6 * n + 2^n.
Note: The number of councilors (n) is not specified in the question, so we cannot provide an exact answer. However, we can provide a formula to calculate the number of ways based on the number of councilors.