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QUESTION: 1

Choose the most appropriate option (a) (b) (c) or (d)

^{4}P_{3} is evaluated as

Solution:

^{4}P_{3}

= 4!/(4!-3!)

= (4!)/1!

= 4*3*2*!

= 24

QUESTION: 2

^{4}P_{4} is equal to

Solution:

QUESTION: 3

is equal to

Solution:

QUESTION: 4

is a symbol equal to

Solution:

0! = 1

QUESTION: 5

In ^{n}P_{r}, n is always

Solution:

QUESTION: 6

In ^{n}P_{r} , the restriction is

Solution:

QUESTION: 7

In^{ n}P_{r} = n (n–1) (n–2) ………………(n–r–1), the number of factor is

Solution:

QUESTION: 8

^{n}P_{r} can also written as

Solution:

QUESTION: 9

If ^{n}P_{4 }= 12 × ^{n}P_{2}, the n is equal to

Solution:

QUESTION: 10

If .^{ n}P_{3} : ^{n}P_{2} = 3 : 1, then n is equal to

Solution:

QUESTION: 11

^{m+n}P_{2} = 56, ^{m–n}P_{2} = 30 then

Solution:

QUESTION: 12

If ^{ 5}P_{r} = 60, then the value of r is

Solution:

QUESTION: 13

If ^{n1+n2}P_{2 }= 132, ^{n1–n2}P_{2} = 30 then,

Solution:

QUESTION: 14

The number of ways the letters of the word COMPUTER can be arranged is

Solution:

Since all letters in the word "COMPUTER" are distinct then the arrangements is 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320

QUESTION: 15

The number of arrangements of the letters in the word FAILURE, so that vowels are always coming together is

Solution:

QUESTION: 16

10 examination papers are arranged in such a way that the best and worst papers never come together. The number of arrangements is

Solution:

No. of ways in which 10 paper can arrange is 10! Ways.

When the best and the worst papers come together, regarding the two as one paper, we have only 9 papers.

These 9 papers can be arranged in 9! Ways.

And two papers can be arranged themselves in 2! Ways.

No. of arrangement when best and worst paper do not come together,

= 10!- 9!.2! = 9!(10-2) = 8.9!.

QUESTION: 17

n articles are arranged in such a way that 2 particular articles never come together. The number of such arrangements is

Solution:

N. particles arrange in a way = n!

if two particles are arranged together then

there is only (n-1) particles

when two particles are together = (n-1)!

and two particles are arranged in= 2! ways

Thus, number of ways in which no particles

are not come together

=n! -2×(n-1)!

=( n-1)!×(n-2)

QUESTION: 18

If 12 school teams are participating in a quiz contest, then the number of ways the first, second and third positions may be won is

Solution:

QUESTION: 19

The sum of all 4 digit number containing the digits 2, 4, 6, 8, without repetitions is

Solution:

QUESTION: 20

The number of 4 digit numbers greater than 5000 can be formed out of the digits 3,4,5,6 and 7(no. digit is repeated). The number of such is

Solution:

QUESTION: 21

4 digit numbers to be formed out of the figures 0, 1, 2, 3, 4 (no digit is repeated) then number of such numbers is

Solution:

QUESTION: 22

The number of ways the letters of the word “Triangle” to be arranged so that the word ’angle’ will be always present is

Solution:

QUESTION: 23

If the letters word ‘Daughter ’ are to be arranged so that vowels occupy the odd places, then number of different words are

Solution:

There are only 3 vowels, a, u, and e, and there are 4 odd places. So

one of the odd spaces must contain a consonant.

We can choose 3 of the 4 odd places to put the vowels in C(4,3) or 4 ways.

For each of those vowel , we can rearrange them in P(3,3) or 3! or 6 ways.

And we can rearrange the 5 consonants in P(5,5) or 5! or 120 ways.

Hence the total number of different words formed are 6*4*120 = 2880

QUESTION: 24

The number of ways in which 7 girls form a ring is

Solution:

QUESTION: 25

The number of ways in which 7 boys sit in a round table so that two particular boys may sit together is

Solution:

No. of ways in which n people can sit in a round table = (n-1)!

So to make them sit together, consider them as a single unit in the sitting arrangement ie. n= 6.

Number of ways so that two particular boys may sit together in the round table:-

(6 -1)!*2

(The two particular boys can also be switched among themselves)

=> 120*2

=> 240

QUESTION: 26

In how many ways can 10 examination papers be arranged so that the best and the worst papers never come together?

Solution:

No. of ways in which 10 paper can arranged is 10! Ways.

When the best and the worst papers come together, regarding the two as one paper, we have only 9 papers.

These 9 papers can be arranged in 9! Ways.

And two papers can be arranged themselves in 2! Ways.

No. of arrangement when best and worst paper do not come together,

= 10! - 9! × 2!

= 9!(10 - 2)

= 8 × 9!

QUESTION: 27

3 ladies and 3 gents can be seated at a round table so that any two and only two of the ladies sit together. The number of ways is

Solution:

QUESTION: 28

The number of ways in which the letters of the word DOGMATIC can be arranged is

Solution:

QUESTION: 29

The number of arrangements of 10 different things taken 4 at a time in which one particular thing always occurs is

Solution:

Number of different things = 10

Numbers of things which are taken at a time = 4

Now, it is said that one particular thing always occur.

So The question reduces to selecting 9 things from the 3 things

This can be done easily by using the concept of combinations

**Number of ways selecting 3 things out of 9 different things such that one particular thing always occur =**

Now, the 4th thing is fixed so number of ways for selecting the 4th thing = 4! = 24

**So, Required number of ways = 84 × 24 = 2016**

QUESTION: 30

The number of permutations of 10 different things taken 4 at a time in which one particular thing never occurs is

Solution:

QUESTION: 31

Mr. X and Mr. Y enter into a railway compartment having six vacant seats. The number of ways in which they can occupy the seats is

Solution:

QUESTION: 32

The number of numbers lying between 100 and 1000 can be formed with the digits 1, 2, 3, 4, 5, 6, 7 is

Solution:

QUESTION: 33

The number of numbers lying between 10 and 1000 can be formed with the digits 2,3,4,0,8,9 is

Solution:

Two digits number (0 can't come on unit place) 5 × 5 = 25

Three digits number 5 × 5 × 4 = 100

Hence the total number of numbers between 10 and 1000 that can be formed using given digits is = 125

QUESTION: 34

In a group of boys the number of arrangement of 4 boys is 12 times the number of arrangements of 2 boys. The number of boys in the group is

Solution:

QUESTION: 35

Solution:

QUESTION: 36

The total number of 9 digit numbers of different digits is

Solution:

QUESTION: 37

The number of ways in which 6 men can be arranged in a row so that the particular 3 men sit together, is

Solution:

QUESTION: 38

There are 5 speakers A, B, C, D and E. The number of ways in which A will speak always before B is

Solution:

QUESTION: 39

There are 10 trains plying between Calcutta and Delhi. The number of ways in which a person can go from Calcutta to Delhi and return by a different train is

Solution:

QUESTION: 40

The number of ways in which 8 sweets of different sizes can be distributed among 8 persons of different ages so that the largest sweat always goes to be younger assuming that each one of then gets a sweet is

Solution:

The number of ways in which 8 sweets of different sizes can be distributed among 8 persons of different ages so that the largest sweat goes to the younger assuming that each one of them gets a sweet.

Since the largest goes to the youngest, that leaves 7 sweets to distribute to 7 people.

^{7}P_{7} = 7! = 5040 ways

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