1. science
2. mathematics

Name_______________________________
Date______
Counting Principle and Permutations
IA1 Per___
Warm-Up:
A movie theater sells 3 sizes of popcorn(small, medium, and large) with 3 choices of toppings(no butter, butter, extra
butter). Draw a tree diagram to determine how many possible ways can a bag of popcorn be purchased?
Tree Diagram
Ordered Pairs
Counting Principle
Sample Space –
Ex. 1: List the sample space for flipping a coin and rolling a die at the same time.
The Counting Principle –
Ex. 2: How many ways can we draw 2 cards from a standard deck of cards without replacement?
Ex. 3: When Mr. Gercke got dressed this morning he walked into his closet to find that he had a choice of 6 pairs of dress
pants, 10 button down shirts, 3 pairs of shoes, and 15 ties. How many possible outfits could he have chose to wear
today?
Ex. 4: Later that same day (after getting dressed), Mr. Gercke found himself at the deli where he had a choice of
breakfasts. He could get eggs cooked either over easy, over medium, or fried. Then he had four choices of cheeses and
three choices of meat. Lastly, he could get his egg sandwich on either a roll or a bagel. How many possible different
sandwiches could Mr. Gercke have gotten this morning?
Permutations – A permutation is an arrangement of objects, without repetition, and order being important. Another
definition of permutation is the number of such arrangements that are possible.
Ex. 5: How many permutations are possible for the letters A, B, C, D, E, and F?
Ex. 6: A chef is preparing a recipe with 7 ingredients. He puts all of one ingredient in a bowl, followed by all of another
ingredient, and so on. How many possible orders are there for placing the 7 ingredients in a bowl in this way?
Ordering subsets of permutations: a permutation of n elements taken r at a time.
Ex. 7: In the Kentucky Derby there are twenty horses in the race. Winning horses are those cross the finish line in first,
second, and third place, commonly called win, place, and show. How many possible arrangements of winning horses are
there at Churchill Downs (the racetrack where the Derby takes place)?
Ex. 8: How many different ways can a chairman, a treasurer, and a secretary be chosen from a group of 12 people?
Ex. 9: The debate team needs to figure out how many different ways they can be lined up for the questions. If there are
seven members of the team, how many ways can they be set up if their team leader, Pat, is always first in line?
Distinct Permutations – Used when there are repeating elements. In order to eliminate this repetition, the total number
of permutations, are divided by the permutations of the repeated elements.
Ex. 10: What is the total number of different nine-letter arrangements that can be formed using the letters in the word
“TELEPHONE”?
Ex. 11: What is the total number of different eleven-letter arrangements that can be formed using the letters in the
word MISSISSIPPI?
Homework:
1. Hawaii wants to issue new license plates consisting of three letters and three numbers, where numbers and letters
can repeat. What is the maximum number of license plates that Hawaii can manufacture?
2. A quiz consists of three true-false questions.
a) How many possible ways are there to answer the questions of this test?
b) Since there is only one way to answer all of the questions correctly, what is the probability that random guessing of all
questions will result in a perfect score?
3. New Mexico issues license plates consisting of letters and numbers where both letters and numbers may be repeated.
Tell how many possible license plates New Mexico may issue when a license plate consists of 2 letters, followed by three
numbers.
4. How many different outfits consisting of a hat, pair of slacks, and a sweater can be made from two hats, three pairs of
slacks, and four sweaters?
5. In how many different ways can a chemistry book, a calculus book, a history book, a poetry book, and a dictionary be
arranged on a shelf so that the chemistry book or the history book appear first?
6. A lottery ticket contains a four-digit number. How many possible four-digit numbers are there when:
a) a digit may appear only once in the number?
b) a digit may appear more than once in the number?
7. What is the total number of different eight-letter arrangements that can be formed using the letters in the word
“PENGUINS”?
8. What is the total number of different twelve-letter arrangements that can be formed using the letters in the word
“PENNSYLVANIA”?