1. A mother, father, and their two children are to line up for a photo. In how many ways can these four family members line up so that the two children are next to each other? Be sure to show your work.

2. Given the set S = {A,B,C,D,E}, how many ways are there to

a. select three letters from S if repetition is allowed?

b. select three letters from S if repetition is not allowed?

c. select one letter from S?

d. select two letters from S such that the first letter is different from the second letter?

e.select three letters from S such that no two letters in the three selected letters are the same?

3. A company has 50 clients and seven employees assigned to them. The manager claims that the most number of clients that each employee will serve is seven. Can the manager's claim be accurate? Explain your answer.

4. Use the generalized definition of lexicographic order to order the following tuples according to lexicographic order.

(5,1,3,2), (4,5,6,1,2), (2,3,4), (7,2), (1,4,7), (5,2,1), (2,5, 3,4,7), (3), (4,2,1), (2,3), (1), (3,4,8)

5. how you arrived at the simplified result following equation below

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