We'll get back our quizzes from last class. Here are solutions:

1. 55. This can be derived either by subtracting the number of 8-bit strings of weight 4 that do not have any 1s in the first two bits (15) from the total number of of 8-bit strings of weight 4 (70), so by summing the number of 8-bit strings of weight 4 with exactly 1 bit in the first two bits (40) with the number containing two 1s in the first two bits (15).
2. 26³ × 10³ = 17,576,000 license plates. We use the product principle because we're making an ordered selection where:

   First position: 26 choices (any letter)
   Second position: 26 choices (any letter)
   Third position: 26 choices (any letter)
   Fourth position: 10 choices (any digit)
   Fifth position: 10 choices (any digit)
   Sixth position: 10 choices (any digit)
   

   So we multiply: 26 × 26 × 26 × 10 × 10 × 10 = 17,576,000
3. Rolling a die and flipping a coin: 6 × 2 = 12 outcomes (product principle) Rolling a die only if the coin shows heads: 1 (tails) + 6 (heads followed by die roll) = 7 outcomes (sum principle) The difference is that in the first scenario, both events always occur, while in the second, the die roll is conditional.

Pairs will present solutions to each of the exercises selected last class. For homework, be sure you understand and can work all the exercises so you are ready for the quiz next Monday.

After discussing the homework, we'll have a quiz on Section 3.2: Combining Outcomes.

After the quiz, we will look at what happens when non-disjoint sets are combined and begin working through the Practice Problems and Additional Exercises in Section 3.3: Non-Disjoint Outcomes.

To encourage more collaboration, let's try pairing up and having each pair prepare the following exercises to present to the class on Thursday. There 11 exercises in total, so we'll formulate a plan to map students to exercises in class.

Period 1B

• Nikita and Jake: Problem 8
• Conrad and Dylan: Problem 2
• Alex and Udval: Problem 4
• Evan and Brendan: Problem 1
• Donavan and Gabriel: Problem 7
• Zach and Luis: Problem 3
• Toby and Rockwell: Problem 5
• Sean and Alessandra: Problem 6
• Grant and (Ellenor and/or Fatima): Problem 9

Period 4B

• Blu and Alex: Problems 1 and 2
• Yuri and Anupama: Problems 4 and 5
• Colin and Rehoboth: Additional Exercise 2
• Hisham and Abi: Problems 6 and 7
• Brayden and Gabe: Problems 8 and 9
• Jack: Problem 3