Name Algebra 1B notes and problems
April 27, 2009Counting problems: practice page 1
Counting problems: practice
First, here’s a review of the counting methods we’ve studied, and some advice on which method tochoose. Your assignment begins on page 2. Homework is to finish whatever you don’t finish in class.
Four counting methods
- Multiplication Principle: When you have two or more decisions to make, the total number of possible combinations is found by multiplying the numbers of options for each decision.
- Factorial (number of ways to put in order): If you have n objects that need to be arranged in anorder, the number of possible orders is “n factorial,” abbreviated n! .
□You can find n! by multiplying all the whole numbers from n down to 1.
□You can find n! on your calculator (n MATH <- 4 ENTER ).
- Permutations (number of ways to select with an order): If you are asked to count the number of ways that routof n objects can be selected with an order, the number of ways is“n permutation r,” abbreviated nPr .
□You can find nPr by multiplying r whole numbers starting from n and counting downward.
□You can find nPr on your calculator (nMATH <- 2 r ENTER ).
- Combinations (number of ways to select without an order): If you are asked to count the number of ways that routof n objects can be selected without an order, the number of ways is “ncombination r,” abbreviated nCr .
□You can find nCr by setting up a fraction in the following way:
on the top, multiply r whole numbers counting downward from n;
on the bottom, multiply r whole numbers counting upward from 1.
□You can find nCr on your calculator (nMATH <- 3 r ENTER).
□You can find nCr from the Pascal’s Triangle number pattern (see April 13).
Choosing which method to use
- If the problem involves making two or more separate decisions and asks about the combined number of possibilities, use the Multiplication Principle.
- If the problem involves a group of objects, and asks a question about how many ways they can be selected or assigned or put in order, use the flowchart below.
Review problems
Directions for problems 1 through 9: Answer these counting questions. For now, do not use the shortcut operations under the MATH key. It’s OK to use your calculator just for arithmetic. Here’s an example of the amount of work you must show: = 35.
1.Hannah is signing up for a summer day camp. She has to pick 4 out of the 9 activities offered by the camp. How many different ways can she make her choices?
2.The principal has to decide the schedule for a half-day. That is, she needs to pick four of the eight class letter blocks, with an order. (For example, the choice could be “C, B,G, F.”) Howmany different schedules are possible?
3.Suppose that you have 12 books that are all different from each other.
a.If you want to arrange all the books in a row on a shelf, in how many different orders could the books be placed?
b.You’ve decided to let a friend borrow 3 of your 12 books. How many different ways could your friend decide which books to borrow?
4.Carly and Jake went to an arcade with 8 different games.
a.Carly decides she wants to play each of the games once. In how many different orders could she decide to play the games?
b.Jake only has enough tokens to play 5 out of the 8 games. In how many different orders could he decide to play the games?
5.The computer in a library children’s room has a password that is just two characters long. Each character can be a capital letter or a number. Here are some examples of possible passwords: Z9, QW, 37, 4T, KK. How many different passwords are possible?
6.A restaurant asks to you fill out a customer satisfaction survey. It looks like this:
quality of food / excellent / good / fair / poortaste of food / excellent / good / fair / poor
variety of food / excellent / good / fair / poor
promptness of service / excellent / good / fair / poor
friendliness of service / excellent / good / fair / poor
restaurant decor / excellent / good / fair / poor
wait time for a table / excellent / good / fair / poor
Suppose that you circle one response (excellent/good/fair/poor) for each item.
How many different ways are there to fill out the survey?
7.The Drama Club has 25 members. It elects 4 different members as president, vicepresident, a secretary, and a treasurer. How many different ways can this be done?
8.The Karate Club has 25 members. It elects 4 of its members as a leadership team. Howmany different ways can this be done?
9.A survey about television watching preferences asks respondents to rate each of the five major television networks (ABC, CBS, CW, Fox, NBC) on a 1-to-10 scale. Which of these is the number of different ways the survey could be completed: 510 or 105 ? Decide which, then find the value.
Directions for the rest of the packet: First identify the calculation that needs to be done, thenfind the answer. You may use the shortcut operations under the MATH key when they apply.
Example of showing enough work: 5P2 = 20 (where the value 20 may be gotten using the shortcut).
10.There are 20 citizens available to serve on a jury. Of them, 12 must be chosen to form a jury. How many ways can the jury be formed?
11.There are 24 basketball teams that could play in a tournament. From them, 16 teams must be chosen for the tournament, with each chosen team given a “seed” of 1st, 2nd, 3rd, …, 16th.
How many different ways can the tournament be formed?
12.Suppose that there are 3 roads connecting Town A to Town B, 4 roads connecting Town B toTown C, and 2 roads connecting Town C to Town D. Here is a picture representing the situation.
How many different ways are there to travel from Town A to Town D? (Assume that the trip goes through each of the towns just once.)
13.Make up a problem just like problem 13, but involving towns W, X, Y, and Z. The answer to theproblem must be “There are 60 different ways to travel from town W to town Z.”
14.A sandwich restaurant offers 9 types of sandwich, 5 types of bread, 3 types of chips, and 6types of drink. How many different meal choices (of sandwich, bread, chips, and drink) arethere?
15.A different restaurant offers a “You Pick Two” deal. There are 8 types of sandwich, 5 types of soup, and 6 types of salad. A meal consists of choices from two different categories.
a.How many different ways are there to order a sandwich and a soup?
b.How many different ways are there to order a sandwich and a salad?
c.How many different ways are there to order a soup and a salad?
d.How many different meals are possible in all?
Hint: Combine your answers to parts a, b, and c.
16.An LCD digit display (used in electronic devices such as digital clocks) has seven segments, eachof which can be turned on or off. For example, todisplay the number 2, five of the segments are turned on and the other two are turned off, as shown in the picture.
Think about all the different possible appearances of this display (not just the ways that represent numbers). How many different displays are possible?
17.For these “make up your own problem” questions, choose counting situations that haven’t already been topic of a problem in class. Don’t just take an idea from another problem you’ve already seen. Best efforts to be creative and original are expected here.
a.Make up your own counting problem whose answer would be 5 · 26 = 130.
b.Make up your own counting problem whose answer would be 5P3 = 5 · 4 · 3 = 60.
c.Make up your own counting problem whose answer would be 5C3 = = 10.
d.Make up your own counting problem whose answer would be 8! = 40320.
e.Make up your own counting problem whose answer would be 28 = 256.