Download - 11X1 T06 03 combinations (2010)
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Combinations
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CombinationsA combination is a set of objects where the order that they are arranged is not important.
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CombinationsA combination is a set of objects where the order that they are arranged is not important.
If we arrange objects in a line, and the order is not important then;
![Page 4: 11X1 T06 03 combinations (2010)](https://reader033.vdocuments.us/reader033/viewer/2022052820/5499d365ac7959222e8b5a02/html5/thumbnails/4.jpg)
CombinationsA combination is a set of objects where the order that they are arranged is not important.
If we arrange objects in a line, and the order is not important then;
A B is the same arrangement as B A
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CombinationsA combination is a set of objects where the order that they are arranged is not important.
If we arrange objects in a line, and the order is not important then;
A B is the same arrangement as B Ae.g. 5 objects, arrange 2 of them
![Page 6: 11X1 T06 03 combinations (2010)](https://reader033.vdocuments.us/reader033/viewer/2022052820/5499d365ac7959222e8b5a02/html5/thumbnails/6.jpg)
CombinationsA combination is a set of objects where the order that they are arranged is not important.
If we arrange objects in a line, and the order is not important then;
A B is the same arrangement as B Ae.g. 5 objects, arrange 2 of them
A B B A C A D A E A
A C B C C B D B E B
A D B D C D D C E C
A E B E C E D E E D
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CombinationsA combination is a set of objects where the order that they are arranged is not important.
If we arrange objects in a line, and the order is not important then;
A B is the same arrangement as B Ae.g. 5 objects, arrange 2 of them
A B B A C A D A E A
A C B C C B D B E B
A D B D C D D C E C
A E B E C E D E E D
20 nsPermutatio 2
5
P
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CombinationsA combination is a set of objects where the order that they are arranged is not important.
If we arrange objects in a line, and the order is not important then;
A B is the same arrangement as B Ae.g. 5 objects, arrange 2 of them
A B B A C A D A E A
A C B C C B D B E B
A D B D C D D C E C
A E B E C E D E E D
20 nsPermutatio 2
5
P
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CombinationsA combination is a set of objects where the order that they are arranged is not important.
If we arrange objects in a line, and the order is not important then;
A B is the same arrangement as B Ae.g. 5 objects, arrange 2 of them
A B B A C A D A E A
A C B C C B D B E B
A D B D C D D C E C
A E B E C E D E E D
20 nsPermutatio 2
5
P
10!2
20 nsCombinatio
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5 objects, arrange 3 of them
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5 objects, arrange 3 of themA B C B A C C A B D A B E A BA B D B A D C A D D A C E A CA B E B A E C A E D A E E A DA C B B C A C B A D B A E B AA C D B C D C B D D B C E B CA C E B C E C B E D B E E B DA D B B D A C D A D C A E C AA D C B D C C D B D C B E C BA D E B D E C D E D C E E C DA E B B E A C E A D E A E D AA E C B E C C E B D E B E D BA E D B E D C E D D E C E D C
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5 objects, arrange 3 of themA B C B A C C A B D A B E A BA B D B A D C A D D A C E A CA B E B A E C A E D A E E A DA C B B C A C B A D B A E B AA C D B C D C B D D B C E B CA C E B C E C B E D B E E B DA D B B D A C D A D C A E C AA D C B D C C D B D C B E C BA D E B D E C D E D C E E C DA E B B E A C E A D E A E D AA E C B E C C E B D E B E D BA E D B E D C E D D E C E D C
60 nsPermutatio 3
5
P
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5 objects, arrange 3 of themA B C B A C C A B D A B E A BA B D B A D C A D D A C E A CA B E B A E C A E D A E E A DA C B B C A C B A D B A E B AA C D B C D C B D D B C E B CA C E B C E C B E D B E E B DA D B B D A C D A D C A E C AA D C B D C C D B D C B E C BA D E B D E C D E D C E E C DA E B B E A C E A D E A E D AA E C B E C C E B D E B E D BA E D B E D C E D D E C E D C
60 nsPermutatio 3
5
P
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5 objects, arrange 3 of themA B C B A C C A B D A B E A BA B D B A D C A D D A C E A CA B E B A E C A E D A E E A DA C B B C A C B A D B A E B AA C D B C D C B D D B C E B CA C E B C E C B E D B E E B DA D B B D A C D A D C A E C AA D C B D C C D B D C B E C BA D E B D E C D E D C E E C DA E B B E A C E A D E A E D AA E C B E C C E B D E B E D BA E D B E D C E D D E C E D C
60 nsPermutatio 3
5
P
10!3
60 nsCombinatio
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If we have n different objects, and we arrange k of them and are not concerned about the order;
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If we have n different objects, and we arrange k of them and are not concerned about the order;
! tsArrangemen ofNumber
kPk
n
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If we have n different objects, and we arrange k of them and are not concerned about the order;
! tsArrangemen ofNumber
kPk
n
!!!
kknn
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If we have n different objects, and we arrange k of them and are not concerned about the order;
! tsArrangemen ofNumber
kPk
n
!!!
kknn
knC
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If we have n different objects, and we arrange k of them and are not concerned about the order;
! tsArrangemen ofNumber
kPk
n
!!!
kknn
knC
e.g. (i) How many ways can 6 numbers be chosen from 45 numbers?
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If we have n different objects, and we arrange k of them and are not concerned about the order;
! tsArrangemen ofNumber
kPk
n
!!!
kknn
knC
e.g. (i) How many ways can 6 numbers be chosen from 45 numbers?
456Ways C
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If we have n different objects, and we arrange k of them and are not concerned about the order;
! tsArrangemen ofNumber
kPk
n
!!!
kknn
knC
e.g. (i) How many ways can 6 numbers be chosen from 45 numbers?
456Ways C
8145060
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If we have n different objects, and we arrange k of them and are not concerned about the order;
! tsArrangemen ofNumber
kPk
n
!!!
kknn
knC
e.g. (i) How many ways can 6 numbers be chosen from 45 numbers?
456Ways C
8145060
Note: at 40 cents per game, $3 258 024 = amount of money you have to spend to guarantee a win in Lotto.
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
511 Committees C
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
511 Committees C With no restrictions, choose 5 people
from 11, gender does not matter
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
511 Committees C With no restrictions, choose 5 people
from 11, gender does not matter 462
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
511 Committees C With no restrictions, choose 5 people
from 11, gender does not matter 462b) the committee contains only males?
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
511 Committees C With no restrictions, choose 5 people
from 11, gender does not matter 462b) the committee contains only males?
57 Committees C
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
511 Committees C With no restrictions, choose 5 people
from 11, gender does not matter 462b) the committee contains only males?
57 Committees C By restricting it to only males, there is
only 7 people to choose from
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
511 Committees C With no restrictions, choose 5 people
from 11, gender does not matter 462b) the committee contains only males?
57 Committees C21
By restricting it to only males, there is only 7 people to choose from
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
511 Committees C With no restrictions, choose 5 people
from 11, gender does not matter 462b) the committee contains only males?
57 Committees C21
By restricting it to only males, there is only 7 people to choose from
c) the committee contains at least one woman?
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
511 Committees C With no restrictions, choose 5 people
from 11, gender does not matter 462b) the committee contains only males?
57 Committees C21
By restricting it to only males, there is only 7 people to choose from
c) the committee contains at least one woman?
21462 Committees
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
511 Committees C With no restrictions, choose 5 people
from 11, gender does not matter 462b) the committee contains only males?
57 Committees C21
By restricting it to only males, there is only 7 people to choose from
c) the committee contains at least one woman?
21462 Committees easier to work out male only and subtract from total number of committees
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(ii) Committees of five people are to be obtained from a group of seven men and four women.
How many committees are possible if;
a) there are no restrictions?
511 Committees C With no restrictions, choose 5 people
from 11, gender does not matter 462b) the committee contains only males?
57 Committees C21
By restricting it to only males, there is only 7 people to choose from
c) the committee contains at least one woman?
21462 Committees 441
easier to work out male only and subtract from total number of committees
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(iii) A hand of five cards is dealt from a regular pack of fifty two cards.
a) What is the number of possible hands?
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(iii) A hand of five cards is dealt from a regular pack of fifty two cards.
a) What is the number of possible hands?
552 Hands C
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(iii) A hand of five cards is dealt from a regular pack of fifty two cards.
a) What is the number of possible hands?
552 Hands C2598960
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(iii) A hand of five cards is dealt from a regular pack of fifty two cards.
a) What is the number of possible hands?
552 Hands C2598960
b) What is the probability of getting “three of a kind”?
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(iii) A hand of five cards is dealt from a regular pack of fifty two cards.
a) What is the number of possible hands?
552 Hands C2598960
b) What is the probability of getting “three of a kind”?
113 Hands C
kind" a ofthree"hasnumber which choose
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(iii) A hand of five cards is dealt from a regular pack of fifty two cards.
a) What is the number of possible hands?
552 Hands C2598960
b) What is the probability of getting “three of a kind”?
113 Hands C
kind" a ofthree"hasnumber which choose
34C
cardsthoseof threechoose
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(iii) A hand of five cards is dealt from a regular pack of fifty two cards.
a) What is the number of possible hands?
552 Hands C2598960
b) What is the probability of getting “three of a kind”?
113 Hands C
kind" a ofthree"hasnumber which choose
34C 2
48C
cardsthoseof threechoose
restthefromcardstworemainingchoose
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(iii) A hand of five cards is dealt from a regular pack of fifty two cards.
a) What is the number of possible hands?
552 Hands C2598960
b) What is the probability of getting “three of a kind”?
113 Hands C
kind" a ofthree"hasnumber which choose
34C 2
48C
cardsthoseof threechoose
restthefromcardstworemainingchoose
58656
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(iii) A hand of five cards is dealt from a regular pack of fifty two cards.
a) What is the number of possible hands?
552 Hands C2598960
b) What is the probability of getting “three of a kind”?
113 Hands C
kind" a ofthree"hasnumber which choose
34C 2
48C
cardsthoseof threechoose
restthefromcardstworemainingchoose
58656
259896058656kind a of three P
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(iii) A hand of five cards is dealt from a regular pack of fifty two cards.
a) What is the number of possible hands?
552 Hands C2598960
b) What is the probability of getting “three of a kind”?
113 Hands C
kind" a ofthree"hasnumber which choose
34C 2
48C
cardsthoseof threechoose
restthefromcardstworemainingchoose
58656
259896058656kind a of three P
291594
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(iii) A hand of five cards is dealt from a regular pack of fifty two cards.
a) What is the number of possible hands?
552 Hands C2598960
b) What is the probability of getting “three of a kind”?
113 Hands C
kind" a ofthree"hasnumber which choose
34C 2
48C
cardsthoseof threechoose
restthefromcardstworemainingchoose
58656
259896058656kind a of three P
291594
(=3.2%)
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A four person team is to be chosen at random from nine women and seven men.
2004 Extension 1 HSC Q2e)
(i) In how many ways can this team be chosen?
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A four person team is to be chosen at random from nine women and seven men.
2004 Extension 1 HSC Q2e)
(i) In how many ways can this team be chosen?
416 Teams C
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A four person team is to be chosen at random from nine women and seven men.
2004 Extension 1 HSC Q2e)
(i) In how many ways can this team be chosen?
416 Teams C With no restrictions, choose 4 people
from 16, gender does not matter
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A four person team is to be chosen at random from nine women and seven men.
2004 Extension 1 HSC Q2e)
(i) In how many ways can this team be chosen?
416 Teams C With no restrictions, choose 4 people
from 16, gender does not matter 1820
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A four person team is to be chosen at random from nine women and seven men.
2004 Extension 1 HSC Q2e)
(i) In how many ways can this team be chosen?
(ii) What is the probability that the team will consist of four women?
416 Teams C With no restrictions, choose 4 people
from 16, gender does not matter 1820
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A four person team is to be chosen at random from nine women and seven men.
2004 Extension 1 HSC Q2e)
(i) In how many ways can this team be chosen?
(ii) What is the probability that the team will consist of four women?
416 Teams C With no restrictions, choose 4 people
from 16, gender does not matter 1820
49 Teams C
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A four person team is to be chosen at random from nine women and seven men.
2004 Extension 1 HSC Q2e)
(i) In how many ways can this team be chosen?
(ii) What is the probability that the team will consist of four women?
416 Teams C With no restrictions, choose 4 people
from 16, gender does not matter 1820
49 Teams C By restricting it to only women, there is
only 9 people to choose from
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A four person team is to be chosen at random from nine women and seven men.
2004 Extension 1 HSC Q2e)
(i) In how many ways can this team be chosen?
(ii) What is the probability that the team will consist of four women?
416 Teams C With no restrictions, choose 4 people
from 16, gender does not matter 1820
49 Teams C126
By restricting it to only women, there is only 9 people to choose from
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A four person team is to be chosen at random from nine women and seven men.
2004 Extension 1 HSC Q2e)
(i) In how many ways can this team be chosen?
(ii) What is the probability that the team will consist of four women?
416 Teams C With no restrictions, choose 4 people
from 16, gender does not matter 1820
49 Teams C126
By restricting it to only women, there is only 9 people to choose from
1820126m women tea4 P
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A four person team is to be chosen at random from nine women and seven men.
2004 Extension 1 HSC Q2e)
(i) In how many ways can this team be chosen?
(ii) What is the probability that the team will consist of four women?
416 Teams C With no restrictions, choose 4 people
from 16, gender does not matter 1820
49 Teams C126
By restricting it to only women, there is only 9 people to choose from
1820126m women tea4 P
1309
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A four person team is to be chosen at random from nine women and seven men.
2004 Extension 1 HSC Q2e)
(i) In how many ways can this team be chosen?
(ii) What is the probability that the team will consist of four women?
416 Teams C With no restrictions, choose 4 people
from 16, gender does not matter 1820
49 Teams C126
By restricting it to only women, there is only 9 people to choose from
1820126m women tea4 P
1309
Exercise 10G; odd(not 19, 27)