Let's go through the problems step by step.

State if each scenario involves a permutation or a combination:

1. A team of 8 basketball players needs to choose a captain and co-captain.

   • This is a permutation because the order matters (captain and co-captain are distinct roles).

2. Rob and Mary are planning trips to 9 countries this year. There are 13 countries they would like to visit. They are deciding which countries to skip.

   • This is a combination because the order in which they skip the countries does not matter.

3. The batting order for seven players on a 12-person team.

   • This is a permutation because the order of the players matters.

4. There are 45 applicants for three Computer Programmer positions.

   • This is a combination because the order in which the positions are filled does not matter.

State if each scenario involves a permutation or a combination, then find the number of possibilities:

5. Castel and Joe are planning trips to three countries this year. There are 7 countries they would like to visit. One trip will be one week long, another two days, and the other two weeks.

   • This is a permutation because the trips are of different durations (order matters).
   • Formula: $P(7, 3) = \frac{7!}{(7-3)!} = \frac{7!}{4!}$
   • Calculation: $P(7, 3) = 7 \times 6 \times 5 = 210$

6. There are 110 people at a meeting. They each shake hands with everyone else. How many handshakes were there?

   • This is a combination because the order of shaking hands doesn't matter.
   • Formula: $C(110, 2) = \frac{110 \times 109}{2}$
   • Calculation: $C(110, 2) = 5995$ handshakes.

7. You are setting the combination on a three-digit lock. You want to use the numbers 123 but don't care what order they are in.

   • This is a combination because the order does not matter.
   • Calculation: There is only 1 possibility because the order doesn’t matter and you already selected the numbers.

8. A group of 25 people are going to run a race. The top 8 finishers advance to the finals.

   • This is a combination because the order does not matter, only who finishes in the top 8.
   • Formula: $C(25, 8) = \frac{25!}{8!(25-8)!}$
   • Calculation: $C(25, 8) = 1,081,575$

Let me know if you'd like to go over more details or explanations.

Here are 5 related questions:

1. How is a permutation different from a combination in real-life scenarios?
2. If you were to choose 5 players from a team of 12, would that be a combination or permutation?
3. Why is the order important when determining handshakes?
4. What is the general formula for calculating permutations of a set?
5. Can you give an example where a combination would be more practical than a permutation?

Tip: A helpful trick is to remember that permutations focus on the arrangement (order matters), while combinations focus on selection (order doesn't matter).