For your college interview, you must wear a tie. You own 3 regular (boring) ties and 5 (cool) bow ties. 36 1. Counting (a) How many choices do you have for your neck-wear? (b) 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? (c) For the rest of your outfit, you have 5 shirts, 4 skirts, 3 pants, and 7 dresses. You want to select either a shirt to wear with a skirt or pants, or just a dress. How many outfits do you have to choose from?

a) 8

b) 15

c) 34

Step-by-step explanation:

Part a) Choices for the neck-wear

Number of regular ties = 3

Number of bow ties = 5

Total number of ties = 3 + 5 = 8

For the neck-wear any tie can be chosen among these 8 available ties. Therefore, the number of choices for the neck-wear would be 8.

Part b)

For this part we need to calculate the number of choices if both neck and bow ties are to be used.

Choosing a regular tie is independent of choosing the bow tie. According to the fundamental rule of counting, if two events are independent of each other, the number of ways of occurrence of both will be equal to the product of their individual occurrences. i.e.

Number of ways to choose both ties = Product of number of ways of choosing each tie

So,

Number of choices for wearing both ties= 3 x 5 = 15 ways

Part c)

Number of shirts = 5

Number of skirts = 4

Number of pants = 3

Number of dresses = 7

Following options are available for the outfits:

Number of ways to choose a pant is 3. Number of ways to chose a skirt or pant is 9.

Using the fundamental rule of counting again:

Number of ways to wear a shirt with skirt or pant = 3 x 9  = 27

Number of ways to chose a dress = 7

Therefore, we have in total 27 +7 = 34 ways of choosing the outfit.