# Add Maths - F5 - (Version 2007) - Permutation and Combination by nklye

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```									               MASTER
in

&

by

NgKL
(M.Ed.,B.Sc.Hons.Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH.)
A. PERMUTATIONS

FACTS:

1.   Multiplication Rule:
If an event A happens in p ways and event B happens in q ways,
then the number of ways that event A happens followed by event B
is p x q.

2.   Permutations:
(a) The number of ways to arrange n unlike objects or events = n!
where n! = n x (n – 1) x (n – 2) ……3 x 2 x 1.

Example: 5! = 5 x 4 x 3 x 2 x 1
= 120

(b) 0! = 1

(c) The number of ways to arrange n objects which have p objects
n!
are alike =
p!

Similarly, if in the n objects have p and q objects which are
alike, then the number of ways to arrange the n objects =
n!
p! q!

(d) The number of ways to arrange r objects taken out from n
n!
unlike objects, n P r =
( n  r )!

5P3           5!       5 x 4 x3 x 2 x1
Example:         =              =                 = 60
( 5  3 )!         2 x1

3.   In a permutations, the order of arrangement of the objects is
important.

Example: The arrangement of ABC and CBA are different!

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Exercise A.1:

1. Calculate the number of ways of       2. Calculate the number of ways of
forming a mixed-double team in           appointing a male class monitor
badminton from 5 male and 4              and a female assistant monitor.
female players.

3. Calculate the number of ways to arrange the following words or digits.

(c) 3, 4, 9, 9                        (d) 2, 3, 5, 10, 11, 25, 25

4. Find the number of different arrangement of the following;

(a) Five letters from the words        (b) Four-digits numbers from the
PROBLEMS, without                      digits 1, 3, 5. 6, 7, 8, 9 if no
repetitions.                           repetition is allowed.

(c) Five boys be arranged in a         (d) 3 chairs be arranged in a
row from a total of 8 boys.            row from 9 different chairs.

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Exercise A.2: Problem Solving I

1. How many 5-digit even numbers         2. How many 4-digit numbers, less
can be arrange from digits 1, 2, 5,      than 5,000 can be formed from the
8, 9 if no repetition is allowed?        digits 3, 4, 5, 6 if no repetition is
allowed?

3. How many different arrangements       4. How many different arrangements
can be formed from the letters of        can be formed from a group of 4
the word TERBILANG if the                boys and 3 girls, if arrangements
arrangement begin with a vowel?          begin with a girl?

Exercise A.3: Problem Solving II

1. How many 4-digit numbers can be formed from digits 0 to 9, if the
numbers are;

(a) odd?                              (b) greater than 8 000?

2. How many 4-letter word codes can be formed from letters of the word
HARMONI if the codes;

(a) contain letter A?                 (b) do not contain any vowel?

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B. COMBINATIONS

FACTS:

1.   The number of combinations of r objects n not alike objects is
nC r
given by

nC
r          n!          n( n  1 )( n  2 ).....( n  r  1 )

( n  r )! r !                   r!

4 C2            4!       4x3x2x1
Example:               =                          6
( 4  2 )!2! (2x1)(2x1)
C0  1
n
2.

Cn  1
n
3.

4.   In a combination, the order of arrangement of objects is not
important.
Example: Arrangement of AB and BA are regarded as one
combination.

Exercise B.1:

1. Find the number of ways of                  2. In how many ways of selecting 3
choosing 4 letters from the word               brands of hand-phones from 7
MASTERY.                                       different brands?

3. A computer club has to select 5             4. Four persons have to be chosen
committee members form 10                      from a group of 4 boys and 3 girls.
students. Find how many ways of                Find how many ways to select
selection?                                     them?

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Exercise B.2: Problem Solving

1. A Parent-Teacher Association has to select 5 teachers from 7 male and 5
female teachers to be represented in the committee. Determine the
numbers of ways to select them to the committee if;

(a) 3 males are to be selected.       (b) At least 2 males to be selected.

2. Four letters are chosen from the word GLORIES. Calculate the number of
selections if

(a) the letter G must be selected.    (b) only one vowel to be selected.

3. A café serves 5 types of food and 3 types of drink for breakfast. A customer
can choose from 1 to 3 types of food or drink for a fixed price. Find the
number of different choices the customer can make with that price if;

(a) only food are to be chosen.       (b) only one type of drink must be
chosen.

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TUTORIAL

1.   Diagram 1 shows seven cards.

U        N       I       F        O       R        M

DIAGRAM 1

A four-letter code is to be formed using four of these cards. Find,
(a) the number of different four-letter codes that can be formed,
(b) the number of different four-letter codes which end with a
consonant.
[4 marks]

(b)……………………………………

2.    A debating team consists of 5 students. These 5 students are chosen
from 4 monitors, 2 assistant monitors and 6 prefects. Calculate the
number of different ways the team can be formed if
(a) there is no restriction,
(b) the team contains only 1 monitor and exactly 3 prefects. [4 marks]

(b)……………………………………
7
Sine positive
(180o )
3.   Diagram 2 shows five cards of different letters.

M       A        T          H        S

DIAGRAM 2

(a) Find the number of possible arrangements, in a row, of all the cards.
(b) Find the number of these arrangements in which the letters A and H
are side by side.                                           [4 marks]

(b)……………………………………

4.   Diagram 3 shows 5 letters and 3 digits.

A    B     C   D    E     6    7       8

DIAGRAM 3

A code is to be formed using those letters and digits. The code must
consists of 3 letters followed by 2 digits.
How many codes can be formed if no letter or digit is repeated in each
code?                                                       [3 marks]