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Permutations and Combinations

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Q 1. In a shop there are five types of ice creams available. A child buys six ice creams.

Statement 1: The number of different ways the child can buy the six ice creams is {tex} ^ { 10 } \mathrm { C } _ { 5 } {/tex}.

Statement 2: The number of different ways the child can buy the six ice creams is equal to the number of different ways of arranging {tex}6 A ^ { \prime } s {/tex} and {tex}4 B ^ { \prime }s {/tex} in a row.

Statement 1 is false, Statement 2 is true

Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1

Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1

Statement 1 is true, Statement 2 is false

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Q 2. From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangements is

less than 500 .

at least 500 but less than 750 .

at least 750 but less than 1000 .

at least 1000

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Q 3. There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

36

66

108

3

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Q 4. Statement {tex} 1 : {/tex} The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is {tex} ^ { 9 } \mathrm { C } _ { 3 } {/tex}.
Statement 2: The number of ways of choosing any 3 places from 9 different places is {tex} ^ { 9 } \mathrm { C } _ { 3 } {/tex} .

Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1

Statement 1 is true, Statement 2 is false

Statement 1 is false, Statement 2 is true

Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1

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Q 5. Let {tex} T _ { n } {/tex} be the number of all possible triangles formed by joining vertices of an {tex} n {/tex} -sided regular polygon. If {tex} T _ { n + 1 } - T _ { n } = 10 {/tex} , then the value of {tex} n {/tex} is

5

10

8

7

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Q 6. Let {tex} A {/tex} and {tex} B {/tex} be two sets containing 2 elements and 4 elements, respectively. The number of subsets of {tex} A \times B {/tex} having 3 or more elements is

220

219

211

256

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Q 7. The sum of the digits in the unit's place of all the {tex}4{/tex} -digit numbers formed by using the numbers {tex} 3,4,5 {/tex} and {tex} 6 , {/tex} without repetition is

432

108

36

18

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Q 8. The number of integers greater than {tex}6000{/tex} that can be formed using the digits {tex} 3,5,6,7 {/tex} and {tex}8{/tex} without repetition is

192

120

72

216

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Q 9. The value of {tex} \underset {r=1}{ \overset {15} \sum} r ^ { 2 } \left( \frac { ^ { 15 } C _ { r } } { ^ { 15 } C _ { r - 1 } } \right) {/tex} is equal to

1240

560

1085

680

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Q 10. If the four letter words (need not be meaningful) are to be formed using the letters from the word 'MEDITERRANEAN' such that the first letter is {tex} R {/tex} and the fourth letter is {tex} E {/tex} , then the total number of all such words is

{tex}\small 110{/tex}

{tex}\small 59{/tex}

{tex}\large \frac { 11 ! } { ( 2 ! ) ^ { 3 } } {/tex}

{tex}\small 56{/tex}

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Q 11. If {tex} \frac { ^{n + 2}C_6 } { ^{n - 2}P_2 } = 11 , {/tex} then {tex} n {/tex} satisfies the equation:

{tex} n ^ { 2 } + n - 110 = 0 {/tex}

{tex} n ^ { 2 } + 2 n - 80 = 0 {/tex}

{tex} n ^ { 2 } + 3 n - 108 = 0 {/tex}

{tex} n ^ { 2 } + 5 n - 84 = 0 {/tex}

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Q 12. Everybody in a room shakes hand with everybody else. The total number of handshakes is equal to {tex} 153 . {/tex} The total number of persons in the room is equal to

18

19

17

16

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Q 13. Number of triangles that can be formed joining the angular points of decagon is

30

20

90

120

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Q 14. A class contains three girls and four boys. Every Saturday, five students go on a picnic, a different group of students is being sent each week. During the picnic, each girl in the group is given a doll by the accompanying teacher. All possible groups of five have gone once. The total number of dolls the girls have got is

21

45

27

24

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Q 15. The number of all the odd divisors of 3600 is

45

4

18

9

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Q 16. There are 'n' numbered seats around a round table. Total number of ways in which {tex} n _ { 1 } \left( n _ { 1 } < n \right) {/tex} persons can sit around the round table is equal to

{tex} ^ { n } C _ { n _ { 1 } } {/tex}

{tex} ^ { n } P _ { n_1} {/tex}

{tex} ^ { n } \mathrm { C } _ { n _ { 1 } - 1 } {/tex}

{tex} ^ { n } P _ { n _ { 1 } - 1 } {/tex}

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Q 17. The total number of three-digit numbers, the sum of whose digits is even, is equal to

450

350

250

325

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Q 18. A teacher takes three children from her class to the zoo at a time as often as she can, but she doesn't take the same set of three children more than once. She finds out that she goes to the zoo 84 times more than a particular child goes to the zoo. Total number of students in her class is equal to

12

14

10

11

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Q 19. Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. The number of ways in which we can place the balls in the boxes (order is not considered in the box) so that no box remains empty is

150

300

200

None of these

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Q 20. Total number of ways of selecting two numbers from the set {tex} \{ 1,2,3,4 , \ldots , 3 n \} , {/tex} so that their sum is divisible by {tex}3{/tex} is equal to

{tex}\large \frac { 2 n ^ { 2 } - n } { 2 } {/tex}

{tex}\large \frac { 3 n ^ { 2 } - n } { 2 } {/tex}

{tex} 2 n ^ { 2 } - n {/tex}

{tex} 3 n ^ { 2 } - n {/tex}

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Q 21. There are five different green dyes, four different blue dyes and three different red dyes. The total number of combinations of dyes that can be chosen taking at least one green and one blue dye is

3255

3720

None of these

The total number of combinations which can be formed of five different green dyes, taking one or more of them is Similarly, by taking one or more of four different red dyes combinations can be formed. The number of combinations which can be formed of three different red dyes, taking none, one or more of them is

Hence, the required number of combinations of dyes

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Q 22. Let and be two lines intersecting at . If are points on and are points on and if none of these coincides with , then the number of triangles formed by these eight points, is

56

55

46

45

If triangle is formed including point the other points must be one from and other point from . Number of triangle formed with ways

When is not included.

Number of triangle formed

Total number of triangles=15+30=45

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Q 23. If eight persons are to address a meeting, then the number of ways in which a specified speaker is to speak before another specified speaker is

2520

20160

40320

None of these

Let be the corresponding speakers.

Without any restriction the eight persons can be arranged among themselves in ways; but the number of ways in which speaks speaks before and the number of ways in which speaks before make up Also, the number of ways in which speaks before is exactly same as the number of ways in which speaks before

So, the required number of ways

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Q 24. The sum of all that can be formed with the digits taken all at a time is

93324

66666

84844

None of these

We have,

Required sum

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Q 25. In an examination of 9 papers a candidate has to pass in more papers, then the number of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful, is

255

256

193

319

The candidate is unsuccessful, if he fails in 9 or 8 or 7 or 6 or 5 papers.

Numbers of ways to be unsuccessful

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