chapter 6

Sheet16.1.4 Permutation of n different objects for given conditions

PRACTICE 6.1.4APRACTICE 6.1.4B

How many 3-digit odd numbers can be formed by using the digits 3, 4, 5, 6 ,7 without repetition? How many 4-digit even numbers can be formed by using 3, 4, 6, 8 ,9 without repetion?

StepsSolutionStepsSolutioni) Form three empty boxesi) Form four empty boxes

ii) Fill the last line with the possibles odd numbers 3P1 ii) Fill the last line with the possibles even numbers 3P1

iii) Fill the rest boxes with the possibles remaining numbers 4P1 3P1 3P1iii) Fill the rest boxes with the possibles remaining numbers 4P1 3P1 2P1 3P1

iv) Use multiplication rule(4P1) X (3P1) X (3P1)iv) Use multiplication rule(4P1) X (3P1) X (2P1) X (3P1)v) Find the numbers of odd numbers those can be formed ( use calculator )36v) Find the numbers of odd numbers those can be formed ( use calculator )72

PRACTICE 6.1.4CPRACTICE 6.1.4D

How many 2-digit even numbers can be formed by using the digits 3, 5, 6, 7, 9 without repetition?How many 5-digit odd numbers can be formed by using the digits 1, 2, 4, 7, 8 without repetition?

StepsSolutionSolution i) Form two empty boxes

ii) Fill the last line with the possible even numbers

iii) Fill the rest boxes with the possibles remaining numbers

iv) Use multiplication ruleX

v) Find the numbers of odd numbers those can be formed4Answer: 48

6.1.3 Permutations of n different objects taken r at a time


Calculate the number of 3-digit that can be form from the digits 3, 4, 7, 8, 9 without repetitionCalculate the number of 5-digit that can be form from the digit 1 to 9 without repetitions

StepsSolutionStepsSolutioni) Form 3 empty lines ( according the pattern digit required )i) Form 5 empty lines ( according the pattern digit required )

ii) Fill in the empty lines with the possibles numbers 5P1 4P1 3P1ii) Fill in the empty lines with the possibles numbers

iii) Use multiplication rule 5P1 X 4P1 X 3P1iii) Use multiplication rule

iv) Find the numbers of odd numbers those can be formed60iv) Find the numbers of odd numbers those can be formed15 120


How many different 4 letter arrangements can be made from the letters of the word 'COMPUTER' without repeating any of the letters?In how many different ways can the letters in the word 'METHOD' be arranged in a row without any repetition?

StepsSolutionSolutioni) Form 4 empty lines ( according the pattern digit required )

ii) Fill in the empty lines with the possibles numbers

iii) Use multiplication rule

iv) Find the numbers of odd numbers those can be formed1680Answer: 720

EXERCISE 6.1.3EXERCISE 6.1.4

1)In how many ways can a 4- digit number be made using 1, 3, 5, 7, 9?1)How many different odd numbers can be formed using the digits 3, 4, 5, 6, and 7. How many of these are less than 50 000?

Answer: 120Answer: 302)There are 10 pairs of trousers in the wardrobe. In how many ways can you arrange 5 of them in a row?2)2 couples and 5 men are arranged in a line. Find the numbers of ways of arranging them if

Answer: 30 240i)the two couples are next to each other and husband and wife are together

3)A small boy has 8 marbles of different colours. In how many ways can he arrange 4 of them in a line?ii)the men are always together.

Answer: 1680Answer: i) 5760 ii) 14 400

4)3)Susan has 5 different water-colour: white, red, yellow, blue and green. How many possible arrangements are there if 3 water - colour are selected and placed in a row

Answer: 606.2COMBINATION

Note:PRACTICE 6.2.1BThe number of combinations of r events from n events = nCrFind the number of ways of selecting 3 cards out of 10 cards.n!( n-r )! r!StepsSolutioni) Identify either to use P or Clook at the phrase of " selections of"n( n-1)(n-2)(n-r+1)r!ii) Identify the n of the questionn =

iii) Identify the r of the questionr = 6.2.1 Combinations of r objects from n objectsiv) Substitute into the formulaePRACTICE 6.2.1Av) use calculator to figure out the answer120There are 15 toys in a box. How many selection of 5 different toys can a child make?

StepsSolution
Acer: Acer:

i) Identify either to use P or Clook at the phrase of " selections of"

ii) Identify the n of the questionn = 15

iii) Identify the r of the questionr = 5

iv) Substitute into the formulae15C5

v) use calculator to figure out the answer3003

6.1PERMUTATION6.1.1The multiplication rule of countingPRACTICE 6.1.1B

PRACTICE 6.1.1AAmin has 5 shirts, 8 ties and 4 pairs of pants. How many dfferent combinations of shirt, tie and pants can he wear?

Osman has 3 different caps, 5 different T-shirts and 4 different pairs of pants. Find the number of different outfits that Osman can put on.StepsSolution

StepsSolution i) Number of shirts

i) Number of caps3ii ) Number of ties

ii ) Number of T-shirts5iii) Number of pants

iii) Number of pants4iv) Multiplication rule

iv) Multiplication rule3 x 5 x 4v) Answer160

v) Answer60

6.1.2: Permutation of n different objectsPRACTICE 6.1.2APRACTICE 6.1.2B

Calculate the number of different arrangements that can be formed if all the letters from the word 'BREAD' are to be arranged.Calculate the number of different arrangements that can be formed if all the letters from the word 'CHAPTER' are to be arranged.

StepsSolutionStepsSolution

i) Number of n5 i) Number of n

ii ) Number of r5ii ) Number of r

iv) Multiplication rule5 x 4 x 3 x 2 x 1iv) Multiplication rule

v) Answer120v) Answer5040

6.2.2 Combinations of r objects chosen from n objects for given condition

PRACTICE 6.2.2A

The school prefects consists of 10 students. These students are chosen from 12 monitors and 10 assistant monitors. Calculate the the number of ways to choose the prefects if the school prefects contains at least 8 monitors.

StepsSolutioni) Form 10 empty lines ( according the pattern digit required )

ii) Determine the pattern can be formedpattern 1: 8 monitors involved so that only 2 assistant monitors can be involved 12C8 X 10C2

pattern 2 : 9 monitors involved so that only 1 assistant monitors can be involved 12C9 X 8C1

pattern 3 : 10 monitors are all involved so that no assistant monitors involved12C10

iii) Notice that, for every single pattern, we need to ADD each other

iv) Answer24 541

PRACTICE 6.2.2B

Parent-Teacher committee, made up of 6 people is chosen from 7 parents, 4 teachers and a principle.In how many ways can the committee be formed if it contains exactly 3 parents

StepsSolutioni) Form 6 empty lines ( according the members of committee required )

ii) Determine the pattern can be formedpattern 1: 3 parents involved so that the rest of 3 will be involved 5 person 7C3 X 5C3

Answer350

PRACTICE 6.2.2C

A committee is made up of 5 people is chosen from 3-men and 6 women. Find the numbers of ways to form the committee if it consists of 2 men only

Solution

Answer: 60

EXERCISES 6.2.1EXERCISES 6.2.2

A school table tennis team is made up by 5 students which is to be chosen from 6 boys and 5 girls. Find the numbers of ways to form the team so that each team consists of 3 boys.

Answer: 200There are 10 students in a club . 4 boys are chosen from the group. In how many ways can they be chosen if the president of the club must always be chosen?

TRY OUT YOURSELFPRACTICE 6.2.1CPRACTICE 6.2.1D

How many ways can a man choose 3 different serving of food from nasi lemak , laksa, prawn noodles, roti canai, pizza, mi rebus and spaghetti?A canteen serves 8 types of food for lunch. A customer can choose from 1 to 3 types of food for a certain price. Find the number of different choices he can make with that price.

StepsSolutionSolutioni) Identify either to use P or C

ii) Identify the n of the question

iii) Identify the r of the question

iv) Substitute into the formulae


Answer : 92

&14PERMUTATION AND COMBINATION

latest 16.1PERMUTATION

6.1.1The multiplication rule of countingPRACTICE 6.1.1B

Note:Amin has 5 shirts, 8 ties and 4 pairs of pants. How many dfferent combinations of shirt, tie and pants can he wear?1Multiplication rule:If an operation A can be performed in m ways, a second operation B can be performed in n ways, then the two successive operations can be performed in m x n waysStepsSolution

i) Number of shirts2Number of permutations of n objects =n!ii ) Number of ties

PRACTICE 6.1.1Aiii) Number of pants

Osman has 3 different caps, 5 different T-shirts and 4 different pairs of pants. Find the number of different outfits that Osman can put on.iv) Multiplication rule

v) Answer160StepsSolution

i) Number of caps3PRACTICE 6.1.1Cii ) Number of T-shirts5A new car model is available with 5 choices of colours and 2 types of engines. How many different variations of this car model are available?iii) Number of pants4

iv) Multiplication rule3 x 5 x 4Solutionv) Answer60

Answer: 10

6.1.2: Permutation of n different objects


Calculate the number of different arrangements that can be formed if all the letters from the word 'BREAD' are to be arranged.Calculate the number of different arrangements that can be formed if all the letters from the word 'CHAPTER' are to be arranged.

StepsSolutionStepsSolution

i) Number of n6 i) Number of n

ii ) Number of r5ii ) Number of r

iv) Multiplication rule5 x 4 x 3 x 2 x 1iv) Multiplication rule

v) Answer120v) Answer5040


Calculate the number of different arrangements that can be formed if all the letters from the word 'FRIEND' are to be arranged.Calculate the number of different arrangements that can be formed if all the letters from the word 'ADVANCED' are to be arranged.

StepsSolutionSOLUTION

i) Number of n

ii ) Number of r

iv) Multiplication rule

v) Answer720

Answer: 40 320


6.1.3 Permutations of n different objects taken r at a time


Calculate the number of 3-digit that can be form from the digits 3, 4, 7, 8, 9 without repetitionCalculate the number of 5-digit that can be form from the digit 1 to 9 without repetitions

StepsSolutionStepsSolutioni) Form 3 empty lines ( according the pattern digit required )i) Form 5 empty lines ( according the pattern digit required )

ii) Fill in the empty lines with the possibles numbers 5P1 4P1 3P1ii) Fill in the empty lines with the possibles numbers

iii) Use multiplication rule 5P1 X 4P1 X 3P1iii) Use multiplication rule

iv) Find the numbers of odd numbers those can be formed60iv) Find the numbers of odd numbers those can be formed15 120


How many different 4 letter arrangements can be made from the letters of the word 'COMPUTER' without repeating any of the letters?In how many different ways can the letters in the word 'METHOD' be arranged in a row without any repetition?

StepsSolutionSolutioni) Form 4 empty lines ( according the pattern digit required )

ii) Fill in the empty lines with the possibles numbers

iii) Use multiplication rule

iv) Find the numbers of odd numbers those can be formed1680Answer: 720

6.1.4 Permutation of n different objects for given conditions


How many 3-digit odd numbers can be formed by using the digits 3, 4, 5, 6 ,7 without repetition? How many 4-digit even numbers can be formed by using 3, 4, 6, 8 ,9 without repetion?

StepsSolutionStepsSolutioni) Form three empty boxesi) Form four empty boxes

ii) Fill the last line with the possibles odd numbers 3P1 ii) Fill the last line with the possibles even numbers 3P1

iii) Fill the rest boxes with the possibles remaining numbers 4P1 3P1 3P1iii) Fill the rest boxes with the possibles remaining numbers 4P1 3P1 2P1 3P1

iv) Use multiplication rule(4P1) X (3P1) X (3P1)iv) Use multiplication rule(4P1) X (3P1) X (2P1) X (3P1)v) Find the numbers of odd numbers those can be formed ( use calculator )36v) Find the numbers of odd numbers those can be formed ( use calculator )72


How many 2-digit even numbers can be formed by using the digits 3, 5, 6, 7, 9 without repetition?How many 5-digit odd numbers can be formed by using the digits 1, 2, 4, 7, 8 without repetition?

StepsSolutionSolution i) Form two empty boxes

ii) Fill the last line with the possible even numbers

iii) Fill the rest boxes with the possibles remaining numbers

iv) Use multiplication ruleX

v) Find the numbers of odd numbers those can be formed4Answer: 48


1)In how many ways can a 4- digit number be made using 1, 3, 5, 7, 9?1)How many different odd numbers can be formed using the digits 3, 4, 5, 6, and 7. How many of these are less than 50 000?

Answer: 120Answer: 302)There are 10 pairs of trousers in the wardrobe. In how many ways can you arrange 5 of them in a row?2)2 couples and 5 men are arranged in a line. Find the numbers of ways of arranging them if

Answer: 30 240i)the two couples are next to each other and husband and wife are together

3)A small boy has 8 marbles of different colours. In how many ways can he arrange 4 of them in a line?ii)the men are always together.

Answer: 1680Answer: i) 5760 ii) 14 4003)Susan has 5 different water-colour: white, red, yellow, blue and green. How many possible arrangements are there if 3 water - colour are selected and placed in a row

Answer: 606.2COMBINATION

Note:PRACTICE 6.2.1BThe number of combinations of r events from n events = nCrFind the number of ways of selecting 3 cards out of 10 cards.n!( n-r )! r!StepsSolutioni) Identify either to use P or Clook at the phrase of " selections of"n( n-1)(n-2)(n-r+1)r!ii) Identify the n of the questionn =

iii) Identify the r of the questionr = 6.2.1 Combinations of r objects from n objectsiv) Substitute into the formulaePRACTICE 6.2.1Av) use calculator to figure out the answer120There are 15 toys in a box. How many selection of 5 different toys can a child make?

StepsSolution
Acer: Acer:

i) Identify either to use P or Clook at the phrase of " selections of"

ii) Identify the n of the questionn = 15

iii) Identify the r of the questionr = 5

iv) Substitute into the formulae15C5



How many ways can a man choose 3 different serving of food from nasi lemak , laksa, prawn noodles, roti canai, pizza, mi rebus and spaghetti?A canteen serves 8 types of food for lunch. A customer can choose from 1 to 3 types of food for a certain price. Find the number of different choices he can make with that price.

StepsSolutionSolutioni) Identify either to use P or C

ii) Identify the n of the question

iii) Identify the r of the question

iv) Substitute into the formulae


Answer : 926.2.2 Combinations of r objects chosen from n objects for given condition

PRACTICE 6.2.2A

The school prefects consists of 10 students. These students are chosen from 12 monitors and 10 assistant monitors. Calculate the the number of ways to choose the prefects if the school prefects contains at least 8 monitors.

StepsSolutioni) Form 10 empty lines ( according the pattern digit required )

ii) Determine the pattern can be formedpattern 1: 8 monitors involved so that only 2 assistant monitors can be involved 12C8 X 10C2

pattern 2 : 9 monitors involved so that only 1 assistant monitors can be involved 12C9 X 8C1

pattern 3 : 10 monitors are all involved so that no assistant monitors involved12C10

iii) Notice that, for every single pattern, we need to ADD each other

iv) Answer24 541

PRACTICE 6.2.2B

Parent-Teacher committee, made up of 6 people is chosen from 7 parents, 4 teachers and a principle.In how many ways can the committee be formed if it contains exactly 3 parents

StepsSolutioni) Form 6 empty lines ( according the members of committee required )

ii) Determine the pattern can be formed 3 parents involved so that the rest of 3 will be involved 5 person 7C3 X 5C3

Answer350

PRACTICE 6.2.2C

A committee is made up of 5 people is chosen from 3-men and 6 women. Find the numbers of ways to form the committee if it consists of 2 men only

Solution

Answer: 60EXERCISES 6.2.1EXERCISES 6.2.2

1A school table tennis team is made up by 5 students which is to be chosen from 6 boys and 5 girls. Find the numbers of ways to form the team so that each team consists of 3 boys.1A team of squash players consists of 6 6 players is chosen from 8 boys and 6 girls. Find the number of ways to form the group if

a) it contains 4 boysb) it contains more boys than girlsAnswer: 2002There are 10 students in a club . 4 boys are chosen from the group. In how many ways can they be chosen if the president of the club must always be chosen?Answer :a) 1050b) 1414

2A 4-digit number is made up from the digit 2, 3, 4, 7, ang 9 . Find the number of ways of getting an even number greater than 3000Answer: 120

Answer: 42

PERMUTATION AND COMBINATION

Sheet3

chapter 6

Documents