Basic Combinatorics – Counting and Permutations
Math Analysis Honors
Example 1 – Counting the Number of Possible Meals
The fixed-price dinner at Yokozuna Japanese Restaurant provides the following choices:
Appetizer: soup or sushi
Main Entrée: Teriyaki chicken, Teriyaki beef, Chicken Katsu, Pork Katsu
Dessert: Green Tea Ice Cream or Mochi ice cream
How many different meals can be ordered if you are only allowed to select one item from each category?
Example 2 – Ordering Pizzas
You are buying a pizza. You have a choice of 3 crusts, 4 cheeses, 5 meat toppings, and 8 vegetable toppings.
a) How many different pizzas with one crust, one cheese, one meat, and one vegetable can you choose?
b) If the pizza crust was not a choice, how many different pizzas could be made?
Example 3 – In how many ways can 7 people be lined up?
Permutations – An ordered arrangement of r objects chosen from n objects.
1) n distinct (different) objects, repetition is allowed for the r objects.
2) n distinct (different) objects, repetition is not allowed for the r objects
3) n non-distinct objects and all objects are used.
Example 4 – The Tennessee license plate consists of three letters of the alphabet followed by three numerical digits (0
through 9). Find the number of different license plates that could be formed
a) if there is no restriction on the letters or digits that be used.
b) If no letter or digit can be repeated.
Example 5 - Count the number of 9-letter “words” (don’t worry about whether they’re in the dictionary) that can be
formed using the letters in each word.
(a) DRAGONFLY
(b) BUTTERFLY
(c) BUMBLEBEE
Distinguishable Permutations
If an n-set contains n1 objects of the
first kind, n2 of the second set, and so
on, with n1  n2  nk  n , then the
number of distinguishable permutations
of the n-set is
n!
n1 !n2 !n3 !   nk !
Permutation Counting Formula
(Distinct/No Repetition)
The number of permutations of n objects
taken r at a time is
n
Pr 
n!
for 0  r  n
n
  r !
Example 6 – Evaluate each expression without a calculator.
(a) 6 P4
(b) 11 P3
(c) n P3
Example 7 – Sixteen actors answer a casting call to try out for roles as dwarfs in a production of Snow White and the
Seven Dwarfs. In how many ways can the director cast the seven roles?
Math Analysis Honors – Worksheet 53
Counting/Permutations
In exercises 1-4, count the number of ways that each procedure can occur.
1.
2.
3.
4.
Line up three people for a photograph.
Prioritize four pending jobs from most to least important.
Arrange five books on a bookshelf from left to right.
Award ribbons for 1st place to 5th place to the top five dogs in a dog show.
5. There are four candidates for homecoming queen and three candidates for king. How many king-queen pairs are
possible?
6. There are three roads form town A to town B and four roads from town B to town C. How many different routes are
there from A to C by way of B?
7. How many 9-letter words can be formed from the letters of the word LOGARITHM?
8. Excluding J, Q, X, and Z, how many 3-letter crossword puzzle entries can be formed that contain no repeated letters?
9. How many distinguishable 11-letter “words” can be formed using the letters in MISSISSIPPI?
10. The 13 members of the East Brainerd Garden Club are electing a President, Vice-President, and Secretary from
among their members. How many different ways can this be done?
11. From among 12 projects under consideration, the mayor must put together a list of 6 projects to submit to the city
council for funding? How many such lists can be formed?
12. How many different license plates begin with two digits, followed by two letters and then three digits if no letters or
digits are repeated?
13. How many different license plates consist of five distinct symbols, either digits or letters?
14. Evaluate each expression without a calculator: a) 6 P2
b) 9 P4