WARM-UP
Determine the total number of way to create a 4-digit number from
the given set of numbers {1, 2, 3, 4, 6, 7, 8}
1) No Restrictions
2) No repeats allowed
3) Must be an even number
4) Must be an even number and no repeats
5) Must be an even number between 2000 and 3999 and no
repeats
WARM-UP - ANSWERS
Determine the total number of way to create a 4-digit number from
the given set of numbers {1, 2, 3, 4, 6, 7, 8}
1) No Restrictions 2401
2) No repeats allowed 840
3) Must be an even number
1372
4) Must be an even number and no repeats
480
5) Must be an even number between 2000 and 3999 and no
repeats 140
PERMUTATION
A permutation of n
objects is a way
of arranging the
objects in order.
Formula
 The number of
permutations of n
objects taken k at a
time is
𝑛!
𝑃 𝑛, 𝑘 =
𝑛−𝑘 !
EXAMPLE
13 workers at JL Graphics Co. enter a contest at
their workplace. 4 different prizes will be
award to the four winners. How many different
ways can first, second, third and fourth prize
be awarded to the 13 workers?
PERMUTATIONS FORMULA #1
• Where k is the number of distinguishable objects, n1 is the
number of indistinguishable objects of the first type,…nk is
the number of indistinguishable of the k th type, and
n1+n2+…+nk=n.
n!
n1!n2 !  nk !
EXAMPLE
How many ways can all of the letters in the word
MISSISSIPPI be arranged in distinguishable
permutations?
PERMUTATIONS FORMULA #2
The number of ordered arrangements of k of n distinguishable
objects with replacement is
𝑛𝑘
EXAMPLE
How many different 4-digit numbers can be
created from the set of digits {1,2,3,4,5,7,9}, if
repeats are allowed?
PERMUTATIONS FORMULA #3
The number of circular permutations of n objects is
𝐶𝑃 𝑛 = 𝑛 − 1 !
EXAMPLE
How many ways can 7 people be seated around
a circular table?
COMBINATIONS
A combinations of n
objects is a way
of arranging the
objects without
regard to order.
Formula
 The number of
combinations of n
objects taken k at a
time is
n!
C (n, k ) 
k!(n  k )!
EXAMPLE #1
A group of 20 students enter a contest with 5 gift
certificates as prizes. Each contestant’s name has
been put into a hat from which 5 names will be drawn
as winners. How many different ways can the five
winners be selected?
EXAMPLE #2
8 students from Mr. Allen’s 2nd period geometry class will be
selected for a special incentive program. The class is
made up of 13 females and 16 males. An additional
restriction has been placed on the selection, the number of
females must equal the number of males. How many
different ways can the 8 students be selected?
EXAMPLE #3
8 students from Mr. Allen’s 2nd period geometry class will be
selected for a special incentive program. The class is
made up of 13 females and 16 males. An additional
restriction has been placed on the selection, the number of
females must be at least 6. How many different ways can
the 8 students be selected?
BLAISE PASCAL
1623 – 1662
French
Mathematician
Physicist
Inventor – Mechanical Calc.
Christian Philosopher
PASCAL’S TRIANGLE
What is the connection to Pascal’s Triangle and Combinations?
Let’s Build it and see…
Pick a row and compare to combinations.
EXAMPLES OF USING PASCAL’S TRIANGLE
Find the combination using Pascal’s Triangle
C(6, 2)
C(3, 1)
C(7, 4)
UNORDERED PARTITIONS
The number of unordered partitions of n objects into m
groups of k each, where n = m  k is
n!
m
m!(k!)
EXAMPLE
How many ways can a professor divide her class of 32 students into 8 groups of 4
students each without regard to order?