GRE : Sample Questions with Solution

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

        a) 32        b) 48       c) 36    d) 60

          Sol :   There are 6 letters in the given word, out of wich there are 3 vowels and 3   

                      consonants.

                     3 vowels can be placed at any of the 3 places  marked 1, 3, 5.

                      No of ways of arranging the vowels = 3P3 = 3! =6

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

                     No of ways these arrangements = 3P3 = 3! = 6

                     Total no of ways = 6*6 =36

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2.  From a group of 7 men and 6 woman five  persons are to be selected to form a committee so that at least 3 men are there on te committee . In how many ways can it be done?

                  a) 564         b) 645    c)  735        d) 756

           Sol :  We may have  (3 men and 2 women ) or (4 men and 1 woman ) or(5 men only)

              Required no of ways = 7C3 * 6C2 + 7C4 * 6C1 +7C5

                                                   = 756

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3. A  box  contains 2 white balls , 3 black balls and 4 red balls . In how many ways can 3 balls be   

   drawn    from the box, if at least one black ball is to be included in the draw?

            a) 32             b) 48          c) 64       d) 96

           Sol: we may have (1 black & 2 non-black ) or (2 black and 1 non black )  or 3 black

                 Required no of ways = 3C1 * 6C2 + 3C2 *   6C1   +  3C3

                                                      = 45+18+1=64

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4. Two dice are thrown together . What  is the probability that the sum of the nos on the two

     faces is divisible by 4 or 6?

          a) 7/18    b) 5/ 18   c) 5/36    d) ½

           Sol :  n(S) = 6 * 6 =36

                     Let E be the event tat the sum of the nos on the two faces is divisible by 4 or 6 Then

                    E = {(1,3) ,(1, 5) , (2, 2), (2, 4) ,(2, 6), (3,1), (3,3) , (3,5), (4,2), (4,4), (5,1), (5,3),(6,2), 

                             (6,6)}

                  n(E) = 14

                   P(E) = n(E)/n(S) = 14/ 36 = 7/18

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5.    A box contains 5 green, 4 yellow and 3 white marbles. Three marbles are drawn at

          random. What is the probability that they are not of the same colour?

         a) 3/44         b)3/55                c) 52/55                  d)41/44

            Sol:  Let S be the sample space. Then n(S) = no of ways of drawing 3 marbles out of 12.

                     = 12C3 = 220

                       Let E be the event of drawing 3 balls of the same colour.

                   Then E = events of drawing (3 balls out of 5) or (3 balls out of 4 ) or (3 balls out of 3)

ð  n(E) = 5C3 +4C3 +3C3 = 5C2 + 4C1 + 1 = 15

ð  P(E) = n(E)/n(S) = 15/220 = 3/44

Required probability = 1-(3/44) =41/44

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