The Multiplicative and Additive Principles

Your wardrobe consists of 5 shirts, 3 pairs of pants, and 17 bow ties. How many different outfits can you make?

For your college interview, you must wear a tie. You own 3 regular (boring) ties and 5 (cool) bow ties.

1. How many choices do you have for your neck-wear?

2. You realize that the interview is for clown college, so you should probably wear both a regular tie and a bow tie. How many choices do you have now?

3. For the rest of your outfit, you have 5 shirts, 4 skirts, 3 pants, and 7 dresses. You want to select…

Your Blu-ray collection consists of 9 comedies and 7 horror movies. Give an example of a question for which the answer is:

1. 16.

2. 63.

Suppose you have sets $A$ and $B$ with $\card{A} = 10$ and $\card{B} = 15$.

1. What is the largest possible value for $\card{A \cap B}$?
2. What is the smallest possible value for $\card{A \cap B}$?
3. What are the possible values for $\card{A \cup B}$?

If $\card{A} = 8$ and $\card{B} = 5$, what is $\card{A \cup B} + \card{A \cap B}$?

Consider all 5 letter “words” made from the letters $a$ through $h$. (Recall, words are just strings of letters, not necessarily actual English words.)

1. How many of these words are there total?

2. How many of these words contain no repeated letters?

3. How many of these words start with the sub-word “aha”?

4. How many of these words either start with “aha” or end with “bah” or both?

5. How many of the words containing…

Suppose a drawer has a dozen brown socks and a dozen black socks (all thrown together).

1. How many socks must be taken out at random to guarantee a match (of any color)?
2. How many socks must be taken out at random to guarantee at least two are black socks?

If there are 38 different time periods during which classes can be scheduled, and there are 677 different classes, how many rooms are required?