A bakery offers 3 types of bread (B1, B2, B3) and 4 types of fillings (F1, F2, F3, F4). A customer chooses one bread and one filling for a sandwich. How many different sandwiches can the customer make? Which is the solution for the problem?

• A. 3! 4!

• B. (3)(4)

• C. 3! + 4!

• D. 3 + 4

A clothing store offers 4 different shirt styles (S1, S2, S3, S4) and 3 different pants styles (P1, P2, P3). A customer wants to buy one shirt and one pair of pants. How many different outfits can the customer choose?

From the problem above, Liam answered there are 12 different outfits the customer can choose. Is he correct?

• A. Yes, because the solution is 4 - 3.

• B. Yes, because the solution is (4)(3).

• C. No, because the solution should be 4! 3!.

• D. No, because the solution should be 4 + 3.

A box contains 2 apples (A1, A2) and 3 bananas (B1, B2, B3). You randomly select one fruit and then another without replacement. Use a tree diagram to visualize the possible outcomes. Which of the following is the correct illustration of the problem?

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A classroom has 5 students: Alice, Bob, Clara, David, and Emily. They want to line up for a photo. What type of permutation is used to calculate the number of arrangements when all 5 students are distinct?

• A. Distinct permutation

• B. Circular permutation

• C. Repeated permutation

• D. Indistinguishable permutation

Which situation illustrates repeated permutation?

• A. Arranging the letters in the word "BALLOON"

• B. Selecting a captain and co-captain from 8 players

• C. Forming a license plate with 3 letters and 3 digits and repetition is allowed

• D. Creating a password for your email

A basketball coach needs to select 3 players from a group of 8 to form a lineup. How many distinct arrangements of 3 players can the coach make? Miguel's solution is 8•7•6 and has an answer of 336 arrangement while Gabriel's solution is 8! and has an answer of 40,320 arrangement. Who is correct and why?

• A. Miguel is correct because he is using the formula for permutations of selecting 3 players from 8 which is appropriate when the order of selection matters.

• B. Gabriel is correct because 8! gives the total number of ways to arrange 8 players, and since the coach only selects 3 players, the full factorial is the correct approach.

• C. Miguel is correct because he is calculating the number of possible ways to arrange 3 players in a specific lineup, which is what the coach needs.

• D. Neither is correct because both Miguel's and Gabriel's methods are incorrect for this problem.

A company manufactures necklaces with 6 beads: 2 red, 2 blue, and 2 yellow beads. How many unique arrangements can the beads be arranged in a straight line?

• A. 90

• B. 120

• C. 180

• D. 720

Four friends are sitting around a circular table: Anna, Ben, Carla, and Dan. How many distinct seating arrangements are possible?

• A. 6

• B. 12

• C. 24

• D. 120

A password consists of 4 characters. Each character can be a letter from the English alphabet (A-Z), and repetition of characters is allowed. How many distinct possible passwords can be created? Daniel's solution is (26)(4) and has an answer of 104 while Gideon's solution is 264 and has an answer of 456,976. Who is correct and why?

• A. Daniel is correct because the total number of possible characters is 26, and there are 4 positions for the characters in the password.

• B. Gideon is correct because for each of the 4 positions, there are 26 possible choices, so the total number of possible passwords is 264.

• C. Both Daniel and Gideon are correct because their methods give the same result.

• D. Daniel nor Gideon is correct because repetition is not allowed.

A teacher chooses 4 out of 10 questions for a quiz and wants to determine the order in which they will appear. How many distinct arrangements are possible?

• A. 210

• B. 5,040

• C. 15,120

• D. 302,400