Multiplication Principle and Tree
Diagram
Tree Diagram
Example:
Imagine that we wish to experimentally
manipulate growth conditions for plants, say the
grass species big bluestem, Andropogon geradi.
We want to grow plants in pots in a greenhouse
at two different levels of fertilizer( low and high )
and four different temperatures (10° 𝐶,
15° 𝐶, 20° 𝐶, 25° 𝐶). If we want three replicates of
each possible combination of fertilizer and
temperature treatment, how many pots will we
need?
Example:
In a medical study, patients are classified
according to whether they have blood type A, B,
AB, or O and also according to whether their
blood pressure is low, normal, or high. In how
many different ways can a patient thus be
classified according to blood type and blood
pressure?
Multiplication Principle
Suppose that an experiment consists of m
ordered tasks. Task 1 has n1 possible outcomes,
task 2 has n2 possible outcomes, …, and task m
has nm possible outcomes. The total number of
possible outcomes of the experiment is
𝑛1 ∙ 𝑛2 ∙ 𝑛3 ⋯ 𝑛𝑚 .
Example
Suppose that after a long day in the greenhouse
you decide to order pizza. You call a local pizza
parlor and learn that there are three choices of
crust and five choices of toppings and that you
can order the pizza with or without cheese. If
you only want one topping, how many different
choices do you have for selecting a pizza?
Example
If an English department schedules four lecture
sections and 12 discussion groups for a course in
modern literature, in how many different ways
can a student choose a lecture section and a
discussion group?
Example
A new-car buyer has the choice of 4 body styles,
3 different engines, and 10 colors. In how many
different ways can a person order one of these
cars?
Permutation
Example
You have three different novels. In how many
different ways can you arrange these books on a
shelf?
Factorial
• 𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2) ⋯ 2 ∙ 1
• By definition, 0! = 1.
Example
Write the formula and calculate 5!, 3!,
5!
.
3!
Example
Suppose that you grow plants in a greenhouse.
To control for spatially varying environmental
conditions, you rearrange the pots every other
day. If you have six pots arranged in a row on a
bench, in how many ways can you arrange the
pots?
Example
A committee of three people, consisting of a
president, a vice president, and a treasurer,
must be chosen from a group of ten. How many
(different) committees can be selected?
Permutation
• P(n,k) or nPk : The number of ways to choose k
objects from n objects and order it.
• nPk = n(n-1)(n-2)…(n-k+1) =
𝑛!
𝑛−𝑘 !
Remark
• nPn=n!
• nP0=1
Example
Calculate
• 5P1
• 5P2
• 5P3
• 5P4
• 5P5
Example
You have just enough time to play four songs out
of ten from your favorite CD. In how many ways
can you program your CD player to play the four
songs?
Example
How many five-letter words with no repeated
letters can you form using the 26 letters of the
alphabet?
Note that a “word” here need not be in the
dictionary.
Combinations
Example
Suppose that you wish to plant five grass species
in a plot. You can choose among twelve different
species. How many choices do you have?
Combination
The number of ways that we choose k
(unordered) objects from n objects.
Example
•
•
•
•
•
•
5C0
5C1
5C2
5C3
5C4
5C5
Example
Compare the quantities in the previous example.
• 5C0
• 5C1
• 5C2
• 5C3
• 5C4
• 5C5
Remark
nCk=nCn-k
That is because selecting k objects from n
objects is equivalent to rejecting n-k object.
Example
A committee of three people must be formed
from a group of 10. How many committees are
there, if no specific tasks are assigned to the
members of the committee?
Problems Selected from HW
• Three people line up for a photograph. How
many different lineups are possible?
Problems Selected from HW
• A group of three students are to be chosen
from a group of 12 students. How many
different groups of three students can be
chosen?
Problems Selected from HW
• Four different awards are to be given to a class
of 10 students. Each student can receive at
most one award. Count the number of ways
these awards can be given out.