Strand 2 Concept 3
K, Grade 1, Grade 2Using manipulatives
Grade 3 Using a diagram
Grade 5, Grade 6Using a systematic approach
HS Using a tree diagram
HS Using multiplication principle
6-HS Determine the possible arrangements of a set of objects (permutations)
HS Determine when to use permutations or combinations
HS Use permutations and combinations to solve problems
Physical Activity to Introduce:
Get 4 volunteers to come to front.
(1) 3 of you get to go to the movies tonight. How many different ways can we select 3 people from
this group of 4?
(2) At the theater, you will all sit together in the same row. Show many different ways 3 of the 4
people can sit at the movies.
DISCUSS:
 What is the difference between (1) and (2)?
 Which had more possibilities?
 Did we NEED a formula?
DISCRETE MATH: Counting Methods
FACTORIALS
Recall that 4! (read “four factorial”) means 4×3×2×1 and n! means n×(n – 1) ×(n – 2) × . . . ×3×2×1.
MULTIPLICATION PRINCIPLE
Suppose that a task involves a sequence of k choices. Let
event can occur and
be the number of ways the first stage or
be the number of ways the second stage or event can occur after the first stage has
occurred. Continuing in this way, let
be the number of ways the kth stage or event can occur after the
first k - 1 stages or events have occurred. Then the total number of different ways the task can occur is:
What does this mean? We can use simple multiplication to answer seemingly complicated questions—
questions that are easily solved with manipulatives when the numbers are small. When the numbers are
large, using such “brute force” methods is not reasonable.
For example, Bobby Bear has two different shirts and three different pairs of pants. How many different
outfits can Bobby Bear make? (Note that an “outfit” consists of one shirt and one pair of pants).
(1) Use manipulatives to solve. Record results.
(2) Use a chart.
(3) Use a diagram.
(4) Use a systematic approach. Make an organized list.
(5) Use the multiplication principle.
Now suppose that Bobby Bear has 15 shirts and 10 pairs of pants. How many outfits can he make?
Now suppose that Bobby Bear is making outfits using 3 shirts, 2 pairs of pants, 4 pairs of socks, and 2
hats. How many different outfits can he make? Note that outfits now consist of one shirt, one pair of
pants, one pair of socks and one hat.
COMBINATIONS
A collection of r objects, WITHOUT regard to ORDER and without repetition, selected from n distinct
objects is called a combination of n objects taken r at a time. The number of such combinations is
denoted by
What does this mean? Combinations count groups of objects when the order of the objects is not important.
For example, suppose that we are having an ice cream party. There are four different flavors of ice cream
and each person can have two scoops in a bowl. How many different ways can we have a bowl of ice
cream (given that the two flavors chosen are not the same)?
Does order matter in this case? Is a scoop of chocolate and a scoop of vanilla the same as a scoop of
vanilla and a scoop of chocolate?
(1) Use a systematic approach. Make an organized list.
(2) Use the formula.
Suppose now, that our party is at Baskin Robbins which has 31 different flavors. How many different
ways can we have our two scoops in a bowl?
(1) Use a systematic approach. Make an organized list.
(2) Use the formula.
PERMUTATIONS
An ORDERED arrangement of r objects, without repetition, selected from n distinct objects is called a
permutation of n objects taken r at a time, and is denoted as
What does this mean? We are now counting the number of ways to arrange or order a set of objects.
For example, suppose that we are having our ice cream party with cones! This time order matters because
chocolate on bottom and vanilla on top is not the same as vanilla on bottom and chocolate on top. NOW
how many ways can we have two scoops of ice cream chosen from 4 different flavors?
(1) Use a systematic approach. Make an organized list.
(2) Use the formula.
(3) Use the multiplication principle.
Suppose now that we are having two-scoop cones at Baskin Robbins. How many different ice cream
cones could we possibly have?
(1) Use a systematic approach. Make an organized list.
(2) Use the formula.
(3) Use the multiplication principle.