Steve interviewed several musicians to be a part of his band...

Question

1. Steve interviewed several musicians to be a part of his band. If he interviewed 5 guitarists, 3 drummers, and 6 vocalists, and the band consists of one guitarist, one drummer, and one vocalist, how many different bands can Steve create?

2. License plates in Florida have the form A12BCD; that is, a letter followed by two digits followed by three more letters.

(a) How many different license plates are possible?

(b) Griff would like a plate that ends in MWS. How many such plates are there?

3. (a) An ice cream shop has vanilla, chocolate, and strawberry flavors. You also have the choice of a sugar cone or a waffle cone. Draw a tree diagram to show the different kinds of double dip cones you could order. (Double dip means two scoops.) Assume that choosing chocolate and then vanilla is different than choosing vanilla and then chocolate. (So, order matters.)

(b) If you must choose two different ice cream flavors for your double dip cone, how many possible different cones could you order?

4. A security keypad uses five digits (0 to 9) in a specific order. How many different keypad patterns are possible if the first three digits must be even and the last digit cannot be zero?

6. The math club needs to select a President, Vice President, Treasurer, and Secretary from their 20 members. How many ways can they do this?

7. The math club wants to choose a committee to plan the back-to-school party. How many 5-person committees can be formed from the set of 20 members?

8. Two freshmen and three sophomores from a sorority will attend a conference. If the sorority has 12 freshmen and 15 sophomores, in how many different ways can the conference attendees are selected?

9. Faculty at a certain university need to create a committee to evaluate the general studies curriculum offered on campus. If there are 25 faculty members and they must select a chair, vice chair, secretary, and 4 additional faculty members to be on the committee, in how many ways can the committee be formed?

10. Sam bought 9 tickets all in the same row to a KC Royals game for himself and his friends next week. His friends are Eric, Sarah, Renee, David, Julie, Derek, Kelsey, and Kyle.

(a) How many different ways can the friends be seated in the row?

(b) Sam has a crush on Sarah, so he wants to make sure his seat is next to hers. How many ways are there to assign seats now?

(c) How many ways are there to assign the seats if Sam wants to ensure that they are all seated boy-girl-boy-girl?

Answer 1

This is an involved question, testing many examples with similar reasoning. They all involve combinations or permutations. A common tactic is multiplying the number of ways to handle different tasks. This is known as the "multiplication principle" or the "rule of product".

1) We choose one guitarist, one drummer, and one vocalist. How many ways can you do each one of those tasks?

Out of five guitarists, we choose one. (five ways)

Out of three drummers, we choose one. (three ways)

Out of six vocalists, we choose one. (six ways)

Multiply each of these answers to find the number of unique bands that can be formed. (5*3*6 = 90 different bands)

6) Select a president, vice president, treasurer, and secretary from 20 members. The key here is that once you've performed one task (e.g. selecting a president) you no longer have that person available to select from in future tasks. So the multiplication would look something like

(20) * (19) * (18) * (17) = 116,280

That is: (20 ways to select one president) * (19 ways to select one vice president from the remaining members) * (18 ways to select one treasurer from the remaining members) * (17 ways to select one secretary from the remaining members)

7) Form a 5-person committee from a set of 20 members.

We do not care about the specific order of the committee members (there is distinction between 'member one' and 'member two' etc.) so we want to use a combination here. The answer would be 20 choose 5, i.e. 20! / (15! * 5!) = 15,504