Question

Suppose a committee wants to decide the service award for last year. There are a total of 6 candidates, Alice, Brigitte, Chris, Dave, Emma, and Frank. Among the six candidates, Alice, Brigitte, and Emma are females and the rest three are males. The committee must decide exactly three people to be awarded, and the nominations must meet all of the following criteria. At least one female must be nominated; Chris and Emma cannot be both nominated; Since Alice and Brigitte work together all the time last year, if one of them is nominated, so is the other; Exactly one of the two people Chris and Dave will be nominated; The nomination cannot be a list containing exactly Alice, Brigitte, and Chris; If Chris cannot get the nomination, neither can Brigitte. Now, as the chair of the committee, how do you decide the nominations?

Answer 1

To decide the nominations, constraints such as having at least one female, Chris and Emma cannot both be nominated, and Alice and Brigitte must be nominated together are considered. Valid combinations are Alice & Brigitte with either Dave or Frank or Emma with either Chris or Frank.

Step-by-step explanation:

The question involves finding valid combinations of nominees based on a set of constraints. Given that at least one female must be nominated, Chris and Emma cannot both be nominated, Alice and Brigitte need to be nominated together, exactly one between Chris and Dave will be nominated, we cannot have just Alice, Brigitte, and Chris together, and if Chris isn't nominated, neither can Brigitte, we can use process of elimination and logical deduction to find the possible nominations.

Let's look at each constraint one-by-one to understand how they impact the nominations:

• At least one female must be nominated, meaning every valid combination must include Alice, Brigitte, and/or Emma.
• Since Alice and Brigitte work together and must be nominated together, they are considered as a single unit for the nominations.
• Chris and Emma cannot both be nominated, so any valid combination cannot have them together.
• Exactly one of Chris and Dave will be nominated, ensuring only one of them can be in a valid combination.
• The nomination cannot be a list containing exactly Alice, Brigitte, and Chris, eliminating that specific combination.
• If Chris is not nominated, Brigitte (and hence Alice) cannot be nominated, meaning any nomination with Dave but without Chris cannot have Alice and Brigitte.

Given these constraints, let's find the valid combinations that can be formed:

• A combination of Alice, Brigitte, and Dave.
• A combination of Alice, Brigitte, and Frank.
• A combination of Emma, Chris, and Dave.
• A combination of Emma, Chris, and Frank.

Since we can't have both Chris and Emma, and Alice and Brigitte must always be together, the last two points constrain the combinations significantly. Also, if we have Dave, we cannot have Alice and Brigitte because that would mean Chris is excluded, which is not allowed.