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M AT H E M AT I C S 1 0
Factorial Notation
n factorial is denoted by n!. It is
the product of all positive
MATHEMATICS 10
integers less than or equal to n.
FACTORIAL NOTATION
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Ms. Christine Magat-Martinez | Teacher III
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Example 1 Example 2
5! 5! + 2!
Answer: 120 Answer: 122
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Example 3 Example 4
8! – 5! 4! 2!
Answer: 40,200 Answer: 48
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Example 5 EXERCISES
6! Evaluate the following expressions.
1. 4! 4. 3! 5!
3! 2. 4! + 2! 5.
5!
3!2!
Answer: 120 3. 7! – 4!
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M AT H E M AT I C S 1 0
Permutation
Permutation is the arrangement of objects in
which order is important. The permutation n
objects taken r at a time is denoted by nPr .
MATHEMATICS 10 𝐧!
PERMUTATION nPr =
𝐧−𝐫 !
; where 0 ≤ r ≤ n
Grade 10 | Pampanga High School
Lourdes, City of San Fernando, Pampanga
Ms. Christine Magat-Martinez | Teacher III
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M AT H E M AT I C S 1 0 EXAMPLES M AT H E M AT I C S 1 0
Permutation Example 1
If n = r, then nPr = n!
Evaluate
5P3
𝐧! 𝐧!
If r = 0, then nP0 = 𝐧−𝟎 ! = 𝐧! = 1
𝐧! 𝐧(𝐧−𝟏)!
If r = 1, then nP1 = = =n
𝐧−𝟏 ! (𝐧−𝟏)! Answer: 60 ways
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Example 2 Example 3
Evaluate Evaluate
4P1 6P6
Answer: 4 ways Answer: 720 ways
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Example 4 Example 5
Evaluate Five persons are to be seated in a
10-seater AUV. In how may ways
6P0 can they be seated?
Answer: 1 way Answer: 30, 240 ways
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Example 6 EXERCISES
Ten hospitals are in need of nurses. If Solve the following problems.
there are four qualified applicants, 1. In how many different ways can a
how many ways can they be assigned? president, vice-president, secretary, and
treasurer be chosen from a class of 15
Answer: 5, 040 ways students?
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EXERCISES
Solve the following problems.
2. A pianist plans to play eight pieces at a
MATHEMATICS 10
recital. How many of these pieces can he
arrange in the program? DISTINGUISHABLE
PERMUTATION
Grade 10 | Pampanga High School
Lourdes, City of San Fernando, Pampanga
Ms. Christine Magat-Martinez | Teacher III
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M AT H E M AT I C S 1 0 M AT H E M AT I C S 1 0
Distinguishable Permutation
Distinguishable Permutation In formula,
𝐧!
The arrangement of objects 𝐏= 𝐧𝟏 !𝐧𝟐 !…𝐧𝐤
, where
n = total no. of objects
with identical objects. n1 = objects of the first kind
n2 = objects of the second kind
nk = objects of the kth kind
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Example 1 Example 2
Find the number of different ways of placing How many ways can we arrange the
15 balls in a row given that 5 are red, 4 are letters in the word COMMITTEE?
green, 3 are yellow, 2 are blue, and one
black
Answer: 37,837,800 ways Answer: 45,360 ways
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Example 3 Example 4
In how many ways can two blue marbles There are 12 people in a dinner gathering. In
and four red marbles be arranged in a how many ways can the host (one of the 12)
row? arrange his guests around a dining table if
a. they can sit on any of the chairs?
b. 3 people insist on sitting beside each other?
Answer: 15 ways
c. 2 people refuse to sit beside each other?
Ms. Christine Magat-Martinez | Teacher III Ms. Christine Magat-Martinez | Teacher III
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ACTIVITY M AT H E M AT I C S 1 0 ACTIVITY M AT H E M AT I C S 1 0
EXERCISES EXERCISES
Solve the problems below. Solve the problems below.
1. How many distinguishable 2. How many distinguishable
permutations are possible with all the permutations are possible with all the
letters of the word ELLIPSES? letters of the word BANANA?
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M AT H E M AT I C S 1 0
Circular Permutation
The arrangement of objects
MATHEMATICS 10
in a circular manner.
CIRCULAR PERMUTATION
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Lourdes, City of San Fernando, Pampanga
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M AT H E M AT I C S 1 0 M AT H E M AT I C S 1 0
Circular Permutation Circular Permutation
The number of circular permutations of The number of permutations of n
n different things is: different things around a key ring and
𝐧! like:
𝐏= = 𝐧−𝟏 !
𝐧𝟏 (𝐧 − 𝟏)!
𝐏=
Ms. Christine Magat-Martinez | Teacher III Ms. Christine Magat-Martinez | Teacher III
𝟐
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Example 1 Example 2
How many ways can 10 different In how many ways can 9 people be
colored toys be arranged in a merry- seated at a round table?
go-around?
Answer: 362,880 ways Answer: 40,320 ways
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Example 3 Example 4
In how many ways can ten keys be In how many ways can eight different
arranged on a key ring? colored beads be arranged on a
bracelet?
Answer: 181,440 ways Answer: 2,520 ways
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EXERCISES EXERCISES
Solve the problems below. 3. There are 12 people in a dinner gathering. In
1. A spinner can be divided into 15 equal parts; how many ways can the host (one of the 12)
how many ways can you arrange 5 colors? arrange his guests around dining table if
2. In how many ways may the vertices of a a. they can sit on any of the chairs?
regular heptagon be named with the letters A, b. 3 people insist on sitting beside each other?
B, C, D, E, F, and G? c. 2 people refuse to sit beside each other?
Ms. Christine Magat-Martinez | Teacher III Ms. Christine Magat-Martinez | Teacher III
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ACTIVITY M AT H E M AT I C S 1 0 ACTIVITY M AT H E M AT I C S 1 0
EXERCISES EXERCISES
Solve the problems below. 5. There are 4 different Mathematics books and 5
4. Five couples want to have their pictures taken. different Science books. In how many ways can the
In how many ways can they arrange themselves books be arranged on a shelf if
a. there are no restrictions?
in a row if b. books of the same subject must be placed together?
a. couples must stay together? c. if they must be placed alternately?
b. they may stand anywhere?
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