# Permutations & Combinations

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There are 2 ways from A to B and 3 ways from B to C. In how many ways can a person reach from A to C?A B C

If a coin is tossed 3 times how many results are possible?H H T H T T H T H T H T H T

Total no of ways is =2*3=6 Why multiplication? The events are independent

Total no of results = 8

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A die is rolled 3 times. Number of outcomes possible is ----(a)1 (b) 3(c) 18 (d) 216 No of results is 6*6*6 = 216

A die is rolled 4 times. Find the no. of distinct sums possible on the top faces.(a)24 (b)1(c) 21 (d) 1296 Minimum sum = 4 Maximum sum = 24 Number of distinct sums possible= 21.

In how many different ways can 10 different varieties of chocolates be distributed to 2 friends? (a) 45 (b) 90 (c) 19 (d) 1024 No. of ways = 2*2*. 10 times = 1024 A coin is rolled n times. Number of outcomes is 2n A die is rolled n times. Number of outcomes is 6n

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In how many ways can ABC seat themselves in 3 places. ABC BCA CAB ACB BAC CBA No. of ways = 6 = 3*2*1=6 Using 1,2,3 how many 3 digit numbers can be formed? Two answers possible. Case 1: With repetition Total = 3*3*3=27 Case 2: Without repetition. Total =3*2*1=6

In how many ways can ABC seat themselves in 3 places, if A& B are identical Red balls and c is a green ball ? RRG RGR GRR RGR RRG GRR No. of ways = 3 How many ways can 3 people be seated in 4 chairs? No. of ways = 4*3*2 =24.How many ways can 4 people be seated in 3 chairs? No. of ways = 4*3*2 =24.

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Permutations

Items distinct

Items non distinct

With repetition

Without repetition

With repetition

Without repetition

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In how many ways can ABCD dance one after another? No of ways = 4*3*2*1 = 24

In how many ways can ABCD be arranged such that A&B are always together? Consider AB as one set. No of ways of arranging becomes = 3!*2! =12 In how many ways can ABCD dance one after another such that A dances before B? No of ways = 24/2 = 12 Why divide by 2?

ACDB BCAB CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA

In how many ways can ABCD dance one after another such that A dances before B and B dances before C? No of ways = 24/3! = 4 Why divide by 6?

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A question paper has 4 questions and each question has 2 choices. In how many a ways can a person answer one or more question?First question cane be answered in 3 ways namely 1a or 1b or 1 not answered. Similarly 2nd , 3rd & 4th questions can be answered in 3 ways each. As answering each question is an independent event, total number is 34 1 = 80 . Why subtract 1?4c

1a 1a 1a 1b 1b 1b 1na 1na 1naNo of questions answered

2a 2b 2 na 2a 2b 2 na 2a 2b 2 naSelection4c 4c 4c 4c 4c 0 1 2 3 4

These 9 ways including one case where no question is answered. Hence 32 1 = 8 .

TOTAL

+ 4c2*22 + 4c3*23 + 4c4*24 = 8+24+32+16 = 80 ways1*2

0 1 2 3 4

1 21 22 23 24

4c 4c 4c 4c 4c

0*2 1*2 2*2 3*2 4*2

0 1 2 3 46

A question paper has 60 questions and each question has 4 choices. In how many a ways can a person answer two or more question? Each question can be answered in 5 ways. Total number of ways of attempting the paper is 560 Not attempting any question is 1 way. Attempting one question is 60* 4 = 240. Total number of ways = 560 1 240 = 560 241

A pool of 10 cops are available for providing security to a VIP. At least 2 cop are needed to provide a security. In how many ways can the security be provided to the VIP? Total number of ways of deploying the cops is 210 , including one way where no cop is deployed. Alotting one cop can be provided in 10 ways.Total number of ways = 210 1 10 = 210 11

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There are 3 apples and 4 bananas . In how many ways can I pick 1 or more fruits? (a) 12 (b) 128 (c) 19 (d) 20 No. of ways = (3+1)(4+1)-1 = 19. Anil wrote down all the possible three-digit numbers with distinct digits on a black board. Of these numbers, Biswas erased all the numbers whose first and last digits were either both even or both odd. How many numbers were left on the board? (1) 450 (2) 360 (3) 400 (4) 320 (5) 540 First Digit Odd : 5*8*5 = 200 First Digit even : 4*8*5 = 160 Total = 360

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How many 4-digit odd numbers are possible such that hundreds digit is 2 more than tens digit? 1) 315 2) 340 3) 350 4) 400 5 )360 The units digit can be filled in 5 ways. The hundreds & tens digits can be filled in 8 ways. The thousands digit can be filled in 9 ways. Total numbers is 9 *8*1* 5 = 360

1000s 100S 10S Units 1 2 3 4 5 6 7 8 9

2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 71 3 5 7 9

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Over a period of 11 days, a vegetable vendor visited exactly 10 villages for selling vegetables. On each day he visited exactly one village, but he did not revisit any village within two days of visiting it. In how many ways could he have visited the villages in the period of 11 days?(1) 10(9)10 (2) 10C2(8)9 (3) 10(8)10 (4) 720(7)8 (5) 90(8)9

In the grid shown alongside, six X's have to be placed such that each row contains at least one X. In how many ways can this be done? (1) 180 (2) 176 (3) 226 (4) 352 (5) 188

10*9*8*8*8*8*8*8*8*8*8 = 90(8)9

Six xs can be placed in 10 cells in 10c6 Let us identify how a row can be kept empty and subtract from total arrangements. Row 1 free. No of arrangement : 1 Row 3 free. No of arrangement : 1 Row 2 free. No of arrangement : 8c6 210 1 - 1 28 = 180

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Among all the four-digit natural numbers divisible by 24, how many have the number 24 in them? (1) 24 (2) 26 (3) 28 (4) 25 (5) None of these Th 2 2 2 2 2 3 6 9 1 4 7 H 4 4 4 4 4 2 2 2 2 2 2 T 0 2 4 7 9 4 4 4 4 4 4 U 0 4 8 2 6 0 0 0 8 8 8 Th 1 1 2 3 3 4 4 5 6 6 7 H 2 8 4 0 6 2 8 4 0 6 2 T 2 2 2 2 2 2 2 2 2 2 2 U 4 4 4 4 4 4 4 4 4 4 411

Th 7 8 9 9

H 8 4 0 6

T 2 2 2 2

U 4 4 4 4

There are 25 values.

The circumference of a circle is divided into 26 equal parts by marking 26 equidistant points on it. Now, using these points as vertices, triangles are drawn such that the circumcentre of each of those triangles lies on one of its sides. How many such triangles can be drawn? (1)156 (2)676 (3)182 (4)650 (5)312

For each diameter there will be 24 such triangles. No of diameters = 13. Totally there will be 512 triangles.12

How many numbers can be formed 4567 can be formed using the digits 3,4,5,6,7 not more than once? (a) 72 (b) 82 (c) 84 (d) 86.4 Digit Nos Starting with 5 Starting with 61000 s 100 s 10 s uni ts

4 digit Nos. 3 digit Nos. 2 digit Nos. 1digit Nos.

5

5 5

5 5 5

5 5 5 5

625 125 25 5 780

8624 24 242 6 613

4 4 4

3 3 3

2 2 2

Starting with 7

4573& 4576 4635,4637,4653, 4657, 4673,4675 4635,4637,4653, 4657,4673,4675

Total no of numbers

How many ways can we give 15 distinct books to 3 students? No. of ways = 15C5* 10C5= 15!/(5!) (5!) (5!)

How many ways the letters of the word GAIN be permuted so that vowels are together?

No of words = 3! * 2! = 12How many ways the letters of the word ALTER be permuted so that vowels are together?

How many ways can we make 3 equal parcels with 15 distinct books?No. of ways =15C5* 10C5 /3!

No of words = 4! * 2! = 48How many ways the letters of the word alert be permuted so that consonants are together?

= 15!/(5!) (5!) (5!) (3!)

No of words = 3! * 3!= 3614

How many ways the letters of the word avert be permuted so that vowels are not together? Total No of words = 5! No of words with vowels being together = 48 No of words with vowels being together = 72 Aliter There are 3 consonants and 2 vowels.Vowels can occupy places marked in 12 . Consonants can be filled in boxes in 6 ways. Totally in 72 ways.

How many ways the letters of the word DAUGHTER be permuted so that no two vowels are together? There are 5 consonants and 3 vowels.

Vowels can occupy places marked in 6*5*4. Consonants can be filled in boxes in 5! ways. Totally in 14400 ways.

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How many nos. >999 and < 4000 can be formed using the digits 0,1,2,3,4 if repetition of digits is allowed? (a) 499 (b) 500 (c ) 375 (d) 376 No of numbers ::: 3*5*5*5 = 375Find the rank of the word COCHIN as arranged in a dictionary? No. of words that begin with C = 5! = 120 No. of words that begin with CC=4! = 24 No. of words that begin with CH=4! = 24 No. of words that begin with CI = 4!= 24 No of words that begin with CN= 4! = 24 The next word is COCHIN Rank of COCHIN is 217.

In how many ways 5 Maths books & 2 physics books can be arranged in a book shelf such that physics books are not together? There are 5 Maths books and 2 Physics books. Physics books can occupy places marked in 6*5. Consonants can be filled in boxes in 5! ways. Totally in 3600 ways.

16

In how many ways 4 Maths books & 3 physics books can be arranged in a book shelf such that physics books are not together? There are 4 Maths books and 3 Physics books. Physics books can occupy places marked in 5*4*3. Consonants can be filled in boxes in 4! ways. Totally in 1440 ways. In how many ways 4 boys & 4 girls sit together such that no two boys & no two girls together? Boys can sit in places marked in in 4*3*2*1. Girls can sit in places marked in in 4*3*2*1. The arrangement can start with a boy or a girl. Therefore totally in 2! 4! 4! = 1152

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A five digit number divisible by 3 is to be formed using the numbers 0 , 1 , 2 , 3 , 4 , 5 without repetition . Find the number of such numbers?For a number to be divisible by 3 sum of the digits should be divisible by 3. Minimum sum possible = 10. Maximum sum possible

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