Section 3.4: The Counting Principle
Objective: To be able to understand and apply the counting
principle.
The Fundamental Counting Principle:
If event 1 occurs M ways and event 2 occurs N ways, then events
1 and 2 occur in succession M*N ways. (This principle can be
extended to more than 2 events.)
Ex. 1 Social Security Numbers
a. How many exist?
b. How many SS numbers start with 3 even numbers?
c. How many SS numbers have no repeats?
Ex. 2 Pin numbers for ATMs
a. How many exist?
b. How many begin and end with an odd number?
c. How many start with the number 4?
Permutations: an ordered arrangement of objects.
(ORDER MATTERS!)
The number of different permutations of n distinct objects is n!
𝑛! = 𝑛 ∙ 𝑛 − 1 ∙ 𝑛 − 2 ∙. . .∙ 3 ∙ 2 ∙ 1
(0! = 1)
Ex. 3 How many different possible standings are there in the
PAC10 assuming that there are no ties?
Ex. 4 How many different ways can you arrange 7 different
baseball cards on a table from left to right?
Permutations of n objects taken r at a time:
𝑛 𝑃𝑟
=
𝑛!
𝑛−𝑟 !
where n > r
Ex. 5 A club has 8 members and selects a president and vice
president. How many ways can this happen?
Ex. 6 A horse race has 7 horses in it. How many different trifecta
bets exist? Trifecta is picking 1st , 2nd and 3rd places in that order.
Distinguishable Permutations: The numbers of permutations of
n objects where 𝑛1 are one type and 𝑛2 are a second type and so
on,
𝑛!
𝑛1 ! ∙ 𝑛2 ! ∙ 𝑛3 ! ∙. . .∙ 𝑛𝑘 !
Ex. 7 There are 7 empty lots in a neighborhood. The builder
wants to build 3 Manor homes, 2 Colonial homes and 2 Ranch
homes. How many ways can these homes be arranged in the 7
lots?
Combinations of n objects taken r at a time:
A combination of r objects taken from a group of n objects
without regard to order is denoted
𝑛 𝐶𝑟 =
𝑛!
𝑟! 𝑛−𝑟 !
where n > r
ORDER DOES NOT MATTER
Ex. 8 How many ways can I select 5 students to go to the office
from our class of 15?
Ex. 9 How many ways can you pick 3 songs to play from a your
music library of 7 songs?
APPLICATIONS OF COUNTING PRINCIPLES:
Ex. 10 Using a standard deck of cards, find the probability of
being dealt 5 diamonds.
Ex. 11 Find the probability of being dealt a 3 of a kind.
Ex. 12 Find the probability of winning a trifecta bet when there
are 6 horses in a horse race.
Ex. 13. Repeat exercise 12 with the bet boxed.