1. The document provides information about permutations including definitions, formulas, and examples of calculating the number of permutations of different arrangements. 2. Permutations refer to the number of ways objects can be ordered or arranged, taking into account the order of the arrangement. 3. Common permutation formulas provided include P(n, n) = n! for permutations of n objects taken n at a time and P(n, r) = n!/(n-r)! for permutations of n objects taken r at a time.

1. ILLUSTRATING
PERMUTATION
QUARTER 3- WEEK 1-2

2. Match each problem on the left to its corresponding answer on
the right. Write the letter of your choice on the space before
each number
Problem Number of Ways
____ 1. In how many ways can six books, A) 12
no two of which are the same,
be arranged in a shelf? B) 24
____ 2. How many permutations are there
for the letters the word ALIVE? C) 120
____ 3. A code is composed of two letters followed
by three digits, using M, C, 1, 2, and 3. D) 600
How many codes can be formed if
repetition is not allowed? E) 720
____ 4. In how many ways can four bikes,
no two of which are the same, F) 800
be displayed in a window?

3. The Factorial Notation ( symbol: ! )
is the process of multiplying
consecutive decreasing whole
numbers from identified number
down to one.
n! = n (n-1) (n-2) (n-3) … (3) (2) (1)
Thus, 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

4. Express in factorial form:
a) 6x5x4x3x2x1
b) 8x7x6x5x4x3x2x1
c) 3x2x1
d) 9x8x7x6x5x4x3x2x1
Determine the value for each expression. Simplify fully
before using a calculator.
a)
10!
5!
d)
12!
8!4!
b)
21!
14!
e)
7!
2!5!
+
7!
4!3!
c)
9!
3!6!
f)
15!
9!6!
+
15!
10!5!

5. What Is a Permutation?
PERMUTATION refers to a
mathematical calculation of the
number of ways a particular set
can be arranged.
PERMUTATION is a word that
describes the number of ways
things can be ordered or arranged.
With permutations, the order of the
arrangement matters.

6. The PERMUTATION of n objects taken n at a time
The number of permutations or arrangements of n
objects taken n at a time in a row is nPn which is
equal to n!
nPn = n!
Example:
1. In how many ways can seven different books
be arranged in a shelf?
2. How many permutations are there for the
letters ALIVE?
3. In how many ways can Aling Rosa arrange 6
potted plants in a row?
4. In how many ways can 5 people arrange
themselves in a row for picture taking?

7. The Permutation of n Objects taken r at a time
nPr =
𝒏!
(𝒏 −𝒓 )!
Example:
1. In how many ways can you place 9
different books on a shelf if there is space
enough for only 5 books?
2. Ten runners join a race.In how
many possible ways cay they be arranged
as first, second and third placers?

8. 3. How many four-digit numbers can be
formed from the numbers 1, 3, 4, 6, 8, and 9 if
repetition of digits is not allowed?
4. Suppose that in a certain association,
there are 12 elected members of the Board of
Directors. In how many ways can a president, a
vice president, a secretary, and a treasurer be
selected from the board?
5. A dress-shop owner has 8 new dresses
that she wants to display in the window. If the
display window has 5 mannequins, in how many
ways can she dress them up?
6. Given the 4-letter word READ. In how many
ways can we arrange its letters, 3 at a time?

9. The number of distinguishable
permutations, P, of n objects where p
objects are alike, q objects are alike, r
objects are alike, and so on, is
P =
𝒏!
𝒑!𝒒!𝒓! …
Example:
1. How many different eight-digit
numbers can be written using the digits 1, 2,
3, 4, 4, 5, 5, and 5?

10. 1. Lisa has three vases of the same kind
and two candle stands of the same kind. In
how many ways can she arrange these items
in a line?
2. Find the number of distinguishable
permutations of the digits of the number
348,838.
3. What is the number of possible
arrangements of nine books on a shelf where
four Algebra books are of the same kind, three
Geometry books are of the same kind, and
two Statistics books are of the same kind?

11. 4. A clothing store has a certain shirt
in four sizes: small, medium, large,
and extra-large. If it has two small,
three medium, six large, and two
extra-large shirts in stock, in how
many ways can these shirts be sold if
each is sold one after the other?
5. How many different nine-digit
numbers can be written using the
following digits: 2,2,2,7,7,8,8,8, and 9?

12. If n objects are arranged in a circle, then
there are
𝒏!
𝒏
or P= (n - 1) ! permutations of
the n objects around the circle.
Example:
1. Ten boy scouts are to be seated
around a camp fire. How many ways
can they be arranged?
2. In how many ways can 4 people
be seated around a circular table?

13. 3. On the buffet there are 7
different appetizers from which
to choose. The appetizers are
arranged on a revolving tray.
How many ways can the
appetizers be organized?
4. How many ways can 8
people be seated at a square
table?

14. Solve for the unknown in each item.
1. P(6, 6) = ___ 6. P(8, r) = 6 720
2. P(7, r) = 840 7. P(8, 3) = ___
3. P(n, 3) = 60 8. P(n, 4) = 3024
4. P(n, 3) = 504 9. P(12, r) = 1320
5. P(10, 5) = ___ 10. P(13, r) = 156