Factorials, Permutations, Tree Diagrams

No work shown is needed. There is 8 questions about factorials, permutations, tree diagrams. I have attached the study guide incase

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1. A designer has designed different tops, pants, and jackets to create outfits. How many different outfits can the models wear if she has designed the following pieces? seven tops, four pants, three jackets

2. You are taking a multiple-choice test that has 7 questions. Each of the questions has 5 answer choices, with one correct answer per question. If you select one of these choices for each question and leave nothing blank, in how many ways can you answer the questions?

3. Specify the number of ways to perform the task described. Give your answers using P(n, r) or C(n, r) notation.

On a biology quiz, a student must match seven terms with their definitions. Assume that the same term cannot be used twice. How many number of ways can you perform the task?

4. Assume you are rolling two dice; the first one is red, and the second is green. Use a systematic listing to determine the number of ways you can roll a total less than 7 on the two dice.

In how many ways can you roll less than 7?

5. An equestrian club has thirteen members. If the club wants to select a president, vice president, and treasurer (all of whom must be different), in how many ways can this be done?

6. Two part question

(a) In a certain state, license plates have two letters followed by four digits. How many such license plates are possible?

(b) If the state department of transportation decides to change the plates to have four letters followed by two digits, how many plates will now be possible?

7. Specify the number of ways to perform the task described. Give your answers using P(n, r) or C(n, r) notation. The key in recognizing whether a problem involves permutations or combinations is deciding whether order is important.

A hot sauce bar has 19 varieties. You are going to select 6 to taste, but you do not care about the order in which you taste them. How many ways can you perform the task?