Name Algebra 1B notes and problems
April 6, 2009 Counting 1: Multiplication Principle page 6
Counting methods (Part 1): The Multiplication Principle
This is the beginning of a unit of the course on counting. Specifically, we will be learning to count things such as the number of different ways that choices can be made, the number of different ways that things can be selected, the number of different ways things can be arranged, and so on.
Here is a very useful counting method called the Multiplication Principle:
When you have two or more choices to make, the total number of possible combinations is found by multiplying the numbers of options for each choice.
Examples using the Multiplication Principle
Q. If someone has 5 shirts and 3 jeans, how many “outfits” (shirt and jeans) can they make?
A. The number of possible outfits is 5 · 3 = 15.
Q. From a list of 10 songs, you are asked to pick your 1st, 2nd, and 3rd favorite songs.
How many different combinations of choices are there?
A. You have 10 choices for the 1st song, 9 choices for the 2nd song (because you can’t choose the same song again), and 8 choices for the 3rd song (do you know why?).
So, the total number of combinations is 10 · 9 · 8 = 720.
Problems using the Multiplication Principle
1. The Class Council is selling T-shirts that come in 3 colors (yellow, blue, white) and
4 sizes (small, medium, large, extra large).
a. Using the multiplication principle, how many different kinds of T-shirts are available?
b. List all of the different kinds of T-shirts that are available. (example: “yellow small”)
2. A card game uses a special deck where each card has a number and a color.
The numbers range from 1 to 12. The colors are red, orange, yellow, green, and blue.
For example, there is a “5 blue” card. If the deck has one card with each of the possible combinations of number and color, how many cards are there in the deck?
3. The table d’hôte (complete meal) at a French restaurant consists of a salad, an entree, and adessert. If there are 2salad choices, 5 entree choices, and 3 dessert choices, how many differentmeals are there?
4. Suppose you flip a coin 4 times. Each time, the coin is either heads (H) or tails (T).
An example of a sequence of 4 coin flips would be: HTTH.
a. List all of the different sequences of 4 coin flips.
b. Answer these questions to figure out how many different coin flip sequences there are.
o How many possible outcomes are there for the first flip? ____
o How many possible outcomes are there for the second flip? ____
o How many possible outcomes are there for the third flip? ____
o How many possible outcomes are there for the fourth flip? ____
o Using the multiplication principle, how many sequences of flips are there? ______
5. If a coin is flipped 6 times, how many different sequences of heads and tails are there?
Shortcut: using an exponent
Sometimes a multiplication problem involves multiplying the same number repeatedly. Ifso, you can calculate the answer using an exponent. For example, in problem 5, instead of multiplying 2·2 · 2 · 2 · 2 · 2 = 64, you could just calculate 26 = 64.
6. If a coin is flipped 20 times, how many different sequences of heads and tails are there? (Calculate using an exponent.)
7. How many different ways could you answer a set of 10 multiple-choice questions,
where each question has answer choices {A, B, C, D, E}?
8. A series of Massachusetts license plates has 4 numbers followed by 2 letters.
Here are some examples of license plates: 1234 AB, 5555 XY, 7070 EE.
Answer these questions to figure out how many different license plates are possible.
o How many possibilities are there for the first number? ____
Hint: The possible digits are 0 through 9.
o How many possibilities are there for the second number? ____
o How many possibilities are there for the third number? ____
o How many possibilities are there for the fourth number? ____
o How many possibilities are there for the first letter? ____
o How many possibilities are there for the second letter? ____
o Now, using the multiplication principle, how many license plates are there? ______
9. A series of New Hampshire license plates has 3 letters followed by 3 numbers.
Here are some examples: ABC 123, SPY 007, ZZZ 789.
How many license plates of this kind are possible?
10. At a bank, every ATM card has a number password called a PIN.
Here are some examples of PIN’s: 7486, 0122, 9999.
Answer these questions to figure out how many different PIN’s are possible.
o How many possibilities are there for the first digit of the PIN? ____
o How many possibilities are there for the second digit of the PIN? ____
o How many possibilities are there for the third digit of the PIN? ____
o How many possibilities are there for the fourth digit of the PIN? ____
o Now, using the multiplication principle, how many sequences are there? ______
11. A web site requires passwords consisting of exactly 5 capital letters.
Here are some examples of passwords: ABCDE, LEXMA, HELLO.
How many different passwords are possible?
12. There are 5 baseball teams in the American League Eastern Division. The teams’ standings for the 2008 season looked like this:
1st place: Tampa Bay
2nd place: Boston
3rd place: New York
4th place: Toronto
5th place: Baltimore
The teams are just starting the 2009 season. How many different standings are possible for this new season? Figure this out by answering the following series of questions.
[Disregard the possibility of teams having a tie. Assume that there will definitely be one team in 1st place, one team in 2nd place, and so on.]
o How many possibilities are there for the 1st place team? ____
o After choosing the 1st place team,
how many possibilities are there for the 2nd place team? ____
o After choosing the 1st and 2nd place teams,
how many possibilities are there for the 3rd place team? ____
o After choosing the 1st, 2nd, and 3rd place teams,
how many possibilities are there for the 4th place team? ____
o After choosing the 1st, 2nd, 3rd, and 4th place teams,
how many possibilities are there for the 5th place team? ____
o Now, using the multiplication principle, how many standings orders are there? ______
Problems 13 and 14 use the same kind of counting strategy as the previous problem.
13. The manager of a softball team needs to put the 9 players on the team into a batting order.
This means: players have to be assigned places from 1st through 9th.
How many different batting orders are possible?
14. The letters ABCDEF can be rearranged in many different orders.
Here are some examples: CFDBAE, DECFAB, BFAEDC.
How many different letter orders are possible?
15. For freshmen at LHS, there are 7 different English teachers and 5 different Earth Science teachers. How many different combinations of English and Earth Science teacher are possible?
16. How many different ways could you answer a set of 25 true-false questions?
17. A web site requires a password consisting of 5 capital letters that must be all different (norepeated letters). How many different passwords are possible? Hint: Think about how many choices there are for the first letter, then for the second letter, and so on.
18. The letters “VERMONT” can be scrambled in many different ways, such as “EOTVRMN”.
How many different ways can the letters be arranged? Hint: Think about how many choices there are for the first letter, then for the second letter, and so on.
19. There are 25 students in a class. Suppose that 3 different prizes will be givento class members by random drawings. Someone who wins one of the prizes is not eligible to win the other prizes. Hint: Think about how many people could be chosen to win the first prize, then the second prize, then the third prize.
20. A school class of 500 students elects a president and a vice-president.
How many combinations of outcomes are possible for this election?
Hint: Handle the offices the same way as the prizes in problem 19.
Check your answers
1. 3 · 4 = 12
2. 12 · 5 = 60
3. 2 · 5 · 3 = 30
4. 2 · 2 · 2 · 2 = 16
5. 2 · 2 · 2 · 2 · 2 · 2 = 64
6. 220 = 1,048,576
7. 510 = 9,765,625
8. 10 · 10 · 10 · 10 · 26 · 26 = 6,760,000
9. 26 · 26 · 26 · 10 · 10 · 10 · = 17,576,000
10. 10 · 10 · 10 · 10 = 10,000
11. 26 · 26 · 26 · 26 · 26 = 11,881,376
12. 5 · 4 · 3 · 2 · 1 = 120
13. 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 362,880
14. 6 · 5 · 4 · 3 · 2 · 1 = 720
15. 7 · 5 = 35
16. 225 = 33,554,432
17. 26 · 25 · 24 · 23 · 22 = 7,893,600
18. 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5,040
19. 25 · 24 · 23 = 13,800
20. 500 · 499 = 249,500