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There is meeting of 20 delegates that is to be held in a hotel. In how many ways these delegates can be seated along a round table, if three particular delegates always seat together?

A.

17! 3!

B.

18! 3!

C.

D.

Answer with explanation

Answer: Option AExplanation

Total 20 persons, 3 always seat together, 17 + 1 =18 delegates can be seated in (18 -1)! Ways = 17!

And now that three can be arranged in 3! Ways.

So, 17! 3! is the correct answer.

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Groups each containing 3 boys are to be formed out of 5 boys. A, B, C, D and E such that no group can contain both C and D together. What is the maximum number of such different groups?

A.

5

B.

6

C.

7

D.

8

Answer with explanation

Answer: Option CExplanation

Maximum number of such different groups = ABC, ABD,ABE, BCE,BDE,CEA,DEA =7.

Alternate method:

Total number of way in which 3 boys can be selected out of 5 is 5C_{3}

Number of ways in which CD comes together = 3 (CDA,CDB,CDE)

Therefore, Required number of ways = 5C_{3} -3

= 10-3 =7.

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Find the number of ways of arranging the letters of the word “MATERIAL” such that all the vowels in the word are to come together?

A.

720

B.

1440

C.

1860

D.

2160

Answer with explanation

Answer: Option BExplanation

n the word, “MATERIAL” there are three vowels A, I, E.

If all the vowels are together, the arrangement is MTRL’AAEI’.

Consider AAEI as one unit. The arrangement is as follows.

M T R L A A E I

The above 5 items can be arranged in 5! ways and AAEI can be arranged among themselves in 4!/2! ways.

Number of required ways of arranging the above letters = 5! * 4!/2!

= (120 * 24)/2 = 1440 ways.

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In how many different ways can the letters of the word ‘OPTICAL’ be arranged so that the vowels always come together?

A.

120

B.

523

C.

560

D.

720

Answer with explanation

Answer: Option DExplanation

The word ‘OPTICAL’ contains 7 different letters.

When the vowels OIA are always together, they can be supposed to form one letter.

Then, we have to arrange the letters PTCL (OIA).

Now, 5 letters can be arranged in 5! = 120 ways.

The vowels (OIA) can be arranged among themselves in 3! = 6 ways.

Required number of ways = (120 x 6) = 720.

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How many words can be formed with or without meaning by taking all the letters from the word SMALL ?

A.

60

B.

120

C.

240

D.

40

Answer with explanation

Answer: Option AExplanation

Number of letters in SMALL is 5, but there are 2 L?s Answer is 5! / 2! = 120 / 2 = 60

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How many different words can be formed from the word DAUGHTER so that ending and beginning letters are consonants?

A.

1440

B.

7200

C.

14400

D.

360

Answer with explanation

Answer: Option CExplanation

Here total letters are 8,3 vowels and 5 consonants. Here 2 consonants can be chosen in ^{5}C_{2} ways

and these 2 consonants can be put it in 2! Ways. The remaining 6 letters can be arranged in 6! Ways.

The words beginning and ending letters with consonant = ^{5}C_{2} *2! *6! = 14400

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In a plane there are totally 8 points (no three points are collinear), how many lines can be drawn ?

A.

28

B.

24

C.

64

D.

56

Answer with explanation

Answer: Option AExplanation

To draw a line we need 2 points. From the given 8 points ,

we need to choose 2 points Number of ways in which this can be done is 18C2= (8 x 7 ) / 2 = 28

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In how many different ways can the letters of the word ‘DETAIL’ be arranged in such a way that the vowels occupy only the odd positions?

A.

26

B.

36

C.

45

D.

52

Answer with explanation

Answer: Option BExplanation

There are 6 letters in the given word, out of which there are 3 vowels and 3 consonants.

Let us mark these positions as under:

(1) (2) (3) (4) (5) (6)

Now, 3 vowels can be placed at any of the three places out 4, marked 1, 3, 5.

Number of ways of arranging the vowels = ^{3}P_{3} = 3! = 6.

Also, the 3 consonants can be arranged at the remaining 3 positions.

Number of ways of these arrangements = ^{3}P_{3} = 3! = 6.

Total number of ways = (6 x 6) = 36.

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