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Mathematics
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Permutation & Combination - 2
Session
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Session Objectives
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Session Objective
1. Combination
2. Circular Permutation
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Combination
Combination Selection
Selection from a, b, c
Select one
Selection Rejectiona , b , c ,b , a , c ,c , a , b ,
Select two
Selection Rejectiona , b, c ,b , c , a , c , a , b ,
No. of ways = 3
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Combination
Select three
Selection Rejectiona, b, c
No. of ways = 1
Number of selection of some from a group.
= Number of rejection of remaining.
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Combination
Number of ways of selecting a group of two student out of four for a trip to Goa.
S 1, S 2, S 3, S 4
Select two
Selection Rejection
S1 S2 S3 S4
S1 S3 S2 S4
S1 S4 S2 S3
S2 S3 S1 S4
S2 S4 S1 S3
S3 S4 S1 S2
6 ways.
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Combination
Number of ways of selecting one group Of two for Goa other for Agra
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Combination
Selection and Arrangement of 3 alphabets from A, B, C, D.
Selection Arrangement Rejection
A, B, C, ABC, ACB, BAC, BCA, CAB, CBA D
A, B, D, ABD, ADB, BAD, BDA, DAB, DBA C
B, C, D, BCD, BDC, CBD, CDB, DBC, DCB A
A, C, D, ACD, ADC, CAD, CDA, DAC, DCA B
43x.3! P
44 4 43
3 1 (4 3)P 4!
x C C C3! 1!3!
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Combination
Number of distinct elements = n (1,2,3, .. n).
Ways of rejecting r = nCr
Ways of rejecting rest (n – r) elements = nCn-r
nCr= nCn-r
Particular selection
Total no. of arrangement = nCr.r! = nPr
1,3, … r elements Arrangement r!
nn r
r
P n!C
r! (n r)!r!
n
n r
n! n!r n r C
(n r)!r!n (n r) !(n r)!
n nr n rC C
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Questions
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Illustrative Problem
There are 5 man and 6 woman. How many way one can select
(a) A committee of 5 person.
(b) A committee of 5 which consist exactly 3 man.
(c) A committee of 5 persons which consist at least 3 man.
Solution :
Man – 5 Woman – 6 Total - 11
115
11!(a) C
6!5!
53
62
(b) Select 3 man C
Select 2 women C
5 6
3 2C C
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Solution Cont.
(c) At least – 3 man
Man – 5 Woman - 6
Composition of CommitteeCase Man Woman
3 2 5C3 x 6C2 = 150
4 1 5C4 x 6C1 = 30
5 0 5C5 x 6C0 = 1
No. of Ways = 150 + 30+ 1 = 181.
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Illustrative Problem
In how many ways, a committee of 4 person Can be selected out of 6 person such that
(a) Mr. C is always there
(b) If A is there B must be there.
(c) A and B never be together.
Solution :
No. of persons - 6 Committee - 4
(a) Available persons – 5 Persons to select - 3
Available persons – 4Persons to select - 2 Ways = 4C2
(b) Case – 1 : ‘ A is there’ – ‘AB’ in Committee.
Ways = 5C3
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Solution Cont.
Case – 2 – ‘A is not there’ – B may /may not be there
Available persons – 5Persons to select - 4 Ways = 5C4
No. of person - 6
Person to Select - 4
(c) ‘AB’ never together = total no. of committee- ‘AB’ always together.
Total no. of committee = 6C4 .
‘AB together in committee’ = 4C2
No. of Ways = 6C4 – 4C2 = 9
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Illustrative Problem
How many straight lines can be drawn through 15 given points. when
(a) No. three are collinear
(b) Only five Points are collinear
Solution :
Through two given point and unique straight line15
2(a) C
(b) 5 point s Collinear5
2C distinct line Considered as one 15 5
2 2Number of Straight line C C 1
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Illustrative ProblemFind the number of 4 digit numbers that can be formed by 3 distinct digits among 1,2,3,4,5
Solution :- No. of digits = 5
31C 4!
2!5 3
3 1
4!No. of ways C C
2!
5 digit Select 3 distinctForm 4 digit nos. using these three
3 digitSelect one which will repeat
Number of digits formed.
5C3
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Illustrative Problem
In how many ways 9 students can be seated both sides of a table having 5 seat on each side (non-distinguishable)
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Circular Permutation
A,B,C,D – to be seated in a circular table
Total line arrangement = 4!
abcd dabc cdab bcdaa
b
c
d
1 circular arrangement
4 linear arrangement
4 linear Arrangement 1 circular arrangement
4 ! linear Arrangement
No. of circular arrangement of n object = (n-1) !
4!or (4 1)! circular arrangement.
4
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Question
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Illustrative Problem
In how many way 4 girls and 5 boys can be seated around a circular table such that
(i) No. two girls sit together
(ii) All girls sit together
(iii) Only two girls sit together
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Solution
(i) Boys – 5 Girls – 4
Boys (5 1)! 4!
Girls 5 4!
B1
B2
B3
B4
B5
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Solution Cont.
(ii) Boys – 5 Girls – 4
B1
B2
B3B4
B5
4!G’s
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Solution Cont.
(iii) Boys – 5 Girls – 4
B1
B2
B3
B4
B5
2!
G2 G1
G3G4
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Invertible Circular Arrangement
Ex :- Garland, Neck less.
Clockwise and anticlockwise arrangement considered as same
ABCD ADCB
(4 1)!2
For n objects.
No. of arrangement = 1
(n 1)!2
AC
D
B
C
D
B
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Question
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Illustrative Problem
I have ten different color stones. In How many way I can make a ring of five stones
Solution :
Stone - 10
Step 2 :- arrange circularly (Invertible)
Step 1 :- choose 510
5C
1 1(5 1)! 4!
2 2 10
5
1Ans : C 4!
2
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Sum of Digits
Find the sum of all three digit numbers formed by 1, 2 and 3
100th place 10th place Unit place
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1
12 12 12
Sum of digits same (12)
Each digits is repeated same no. of times=2=(3-1) !
All digits comes equal no. of times
Sum of digits in each column = (1+2+3) x 2! = 12
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Sum of Digits
100th place 10th place Unit place
a b c
12 12 12
= 100a +10b +c
= 100x12 +10x12 + 12
= 12 (102 + 10 + 1)
= 12 x 111 = 1332
(sum of all digits) x (No. of repetition in a particular column) x (No. of 1’s as number of digits present in the number
Sum of all numbers =
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Class Test
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Class Exercise - 1
In how many ways can 5 boys and5 girls be seated in a row so that no2 girls are together and at least2 boys are together?
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Solution
First the boys can be seated in 5p5 = 5! ways.
Each arrangement will create six gaps:
__ B __ B __ B __ B __ B
If the girls are seated in the gaps, no 2 girls will be together. Girls can be seated in the gaps in 6p5 = 6! ways. But if the girls are in the first five or the last five gaps, no 2 boys will be together. So the girls can be seated in 6! – (5! + 5!) = 6! – 2 × 5! = 4 × 5!
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Solution contd..
Thus, total arrangements possible are 4 x (5!)2
= 4 × 120 × 120= 57600
Note: Under the given condition, more than 2 boys cannot sit together.
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Class Exercise - 2
Find the sum of all numbers formedusing the digits 0, 2, 4, 7.
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SolutionRequired sum is 3 24 .13.1111 4 .13.111
n n 2n 1 n 2
10 1 10 1n .S. n .S.
10 1 10 1
24 .13 4444 111
= 16 × 13 × 4333
= 208 × 4333
= 901264
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Class Exercise - 3
If all the letters of the word ‘SAHARA’are arranged as in the dictionary, whatis the 100th word?
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SolutionArranging the letters alphabetically, we have A, A, A, H, R, S.
5!60
2!Number of words starting with A:
Number of words starting with H:5!
203!
Number of words starting with R:5!
203!
Thus, the last word starting with R will be the100th word. This is clearly RSHAAA.
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Class Exercise - 4How many numbers can be formedusing the digits 3, 4, 5, 6, 5, 4, 3such that the even digits occupythe even places?
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Solution
Even digits are 4, 6, 4.
These can be arranged in the even places in
ways = 3 ways.3!2!
Thus, the total number of ways = 3 × 6 = 18 ways.
The remaining digits: 3, 5, 5, 3 can be arranged
in the remaining places in ways.4!
62! 2!
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Class Exercise - 5Ten couples are to be seated arounda table. In how many ways can theybe seated so that no two neighboursare of the same gender?
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Solution
Let all the members of one gender be seated around the table. This can be done in (10 – 1)! ways. Once one gender is seated, arrangement of other gender is no longer a problem of circular permutation (since the seats can be identified). Thus, the second gender can be seated in 10! ways.
Thus, total ways = 9! × 10!
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Class Exercise - 6In how many ways can 15 delegatesbe seated around a pentagonal tablehaving 3 chairs at each edge?
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Solution
12
3
4
5
6789
10
11
12 13
14 15 15
I2
3
4
5678
9
10
1112
1314
If we consider the problem as one of circular permutations, the answer is (15 – 1)! = 14!
But we are considering the above two arrangements as same while they are clearly different. All that has been done is that all delegates have shifted one position. One move shift will also give a new arrangement.
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Solution contd..
Thus, we are counting three different arrangements as one.
Thus, number of actual arrangements possible = 3 × 14!
But after three shifts, the arrangement will be
which is identical to the original arrangement.
1314
15
1
2
3456
7
8
910
1112
or , where 5 is the number of sides of regular polygon.15!5
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Class Exercise - 7Prove that the product of r consecutiveintegers is divisible by r!
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SolutionLet the r consecutive integers be
(n + 1), (n + 2), (n + 3), ..., (n + r)
Product = (n + 1)(n + 2)(n + 3) ... (n + r)
n! n 1 n 2 n 3 ... n r
n!
rr
n r !n P
n!
But n+rPr is an integer.
Thus, the product of r consecutive integers is divisible by r!(Proved)
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Class Exercise - 8If
find the values of n and r.
n n n nr r 1 r r 1p p and C C ,
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Solutionn n
r r 1p p
n! n!
n r ! n r 1 !
n r 1
n nr r 1Also C C
n! n!
n r ! r! n r 1 ! r 1 !
n! n!
n r ! r! n r 1 ! r 1 !
r 2, n 3
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Class Exercise - 9A person wishes to make up as manyparties as he can out of his 18 friendssuch that each party consists of thesame numbers of persons. How manyfriends should he invite?
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Solution
Let the person invite r friends. This canbe done in 18Cr ways. To maximise thenumber of parties, we have to take thelargest value of 18Cr. When n is even,nCr will be maximum when r= n/2.
Thus, he should invite 18/2 = 9 friends.
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Class Exercise - 10In how many ways can a cricket teamof 5 batsmen, 3 all-rounders, 2 bowlersand 1 wicket keeper be selected from19 players including 7 batsmen,6 all-rounders, 3 bowlers and3 wicket keepers?
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Solution
The batsmen can be selected in 7C5 = 21 ways.
The all-rounders can be selected in 6C3 = 20 ways.
The bowlers can be selected in 3C2 = 3 ways.
The wicketkeeper can be selected in 3C1 = 3 ways.
Thus, the total ways of selecting the team
= 21 × 20 × 3 × 3 = 3780 ways.
Note: 7 75 2
7.6C C 21
2.1
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Class Exercise - 11In how many ways can we select oneor more items out of a, a, a, b, c, d, e?
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Solution
We can select ‘a’s in 0 or 1 or 2 or 3, i.e. in 4 ways.
We can select ‘b’ in 2 ways, i.e. either we select it or we do not select it and so on.
The required number of ways in which we can select one or more items is
(3 + 1)(1 + 1)(1 + 1)(1 + 1)(1 + 1) – 1
44.2 1 63 ways.
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Class Exercise - 12
In how many ways can we divide10 persons (i) into groups of 5 each,(ii) (ii) into groups of 4, 4 and 2?
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Solution
105C
(i)2!
We have divided by 2!, because if we interchange persons in group one, with persons in group two, the division is not different, i.e.
group 1 group 2 group 2 group 1
abdfj ceghi ceghi abdfj
10 6 24 4 2C . C . C 10! 1
(ii)2! 4! 4! 2! 2!
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Thank you