
question_answer1) The value of \[{{2}^{n}}\,\{1.3.5.....(2n3)\,(2n1)\}\] is
A) \[\frac{(2n)\,!}{n\,!}\]
B) \[\frac{(2n)\,!}{{{2}^{n}}}\]
C) \[{{2}^{ni}}\]
D) None of these
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question_answer2) A question paper is divided into two parts A and B and each part contains 5 questions. The number of ways in which a candidate can answer 6 questions selecting at least two questions from each part is [Roorkee 1980]
A) 80
B) 100
C) 200
D) None of these
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question_answer3) How many numbers lying between 10 and 1000 can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 (repetition is allowed)
A) 1024
B) 810
C) 2346
D) None of these
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question_answer4) The number of ways in which the letters of the word TRIANGLE can be arranged such that two vowels do not occur together is
A) 1200
B) 2400
C) 14400
D) None of these
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question_answer5) There are four balls of different colours and four boxes of colours same as those of the balls. The number of ways in which the balls, one in each box, could be placed such that a ball does not go to box of its own colour is [IIT 1992]
A) 8
B) 7
C) 9
D) None of these
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question_answer6) If \[^{56}{{P}_{r+6}}{{:}^{54}}{{P}_{r+3}}=30800:1\], then \[r=\] [Roorkee 1983; Kurukshetra CEE 1998]
A) 31
B) 41
C) 51
D) None of these
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question_answer7) Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated is [IIT 1980; MNR 1998, 99; DCE 2001]
A) 69760
B) 30240
C) 99748
D) None of these
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question_answer8) The number of ways of dividing 52 cards amongst four players so that three players have 17 cards each and the fourth player just one card, is [IIT 1979]
A) \[\frac{52\ !}{{{(17\ !)}^{3}}}\]
B) \[52\ !\]
C) \[\frac{52\ !}{17\ !}\]
D) None of these
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question_answer9) The number of ways in which the letters of the word ARRANGE can be arranged such that both R do not come together is [MP PET 1993]
A) 360
B) 900
C) 1260
D) 1620
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question_answer10) A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw [IIT 1986; DCE 1994]
A) 64
B) 45
C) 46
D) None of these
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question_answer11) \[m\] men and n women are to be seated in a row so that no two women sit together. If \[m>n\], then the number of ways in which they can be seated is [IIT 1983]
A) \[\frac{m\ !\ (m+1)\ !}{(mn+1)\ !}\]
B) \[\frac{m\ !\ (m1)\ !}{(mn+1)\ !}\]
C) \[\frac{(m1)\ !\ (m+1)\ !}{(mn+1)\ !}\]
D) None of these
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question_answer12) A five digit number divisible by 3 has to formed using the numerals 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which this can be done is [IIT 1989; AIEEE 2002]
A) 216
B) 240
C) 600
D) 3125
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question_answer13) In a certain test there are \[n\] questions. In the test \[{{2}^{ni}}\] students gave wrong answers to at least \[i\] questions, where \[i=1,\ 2,\ ......n\]. If the total number of wrong answers given is 2047, then \[n\] is equal to
A) 10
B) 11
C) 12
D) 13
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question_answer14) The number of times the digit 3 will be written when listing the integers from 1 to 1000 is
A) 269
B) 300
C) 271
D) 302
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question_answer15) Ten persons, amongst whom are A, B and C to speak at a function. The number of ways in which it can be done if A wants to speak before B and B wants to speak before C is
A) \[\frac{10\ !}{6}\]
B) \[3\ !\ 7\ !\]
C) \[^{10}{{P}_{3}}\ .\ 7\ !\]
D) None of these
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question_answer16) The number of ways in which an examiner can assign 30 marks to 8 questions, awarding not less than 2 marks to any question is
A) \[^{21}{{C}_{7}}\]
B) \[^{30}{{C}_{16}}\]
C) \[^{21}{{C}_{16}}\]
D) None of these
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question_answer17) How many words can be made out from the letters of the word INDEPENDENCE, in which vowels always come together [Roorkee 1989]
A) 16800
B) 16630
C) 1663200
D) None of these
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question_answer18) Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. In how many ways can we place the balls so that no box remains empty [IIT 1981]
A) 50
B) 100
C) 150
D) 200
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question_answer19) In how many ways can a committee be formed of 5 members from 6 men and 4 women if the committee has at least one woman [RPET 1987; IIT 1968; Pb. CET 2003]
A) 186
B) 246
C) 252
D) None of these
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question_answer20) How many words can be made from the letters of the word BHARAT in which B and H never come together [IIT 1977]
A) 360
B) 300
C) 240
D) 120
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question_answer21) There are 10 persons named \[A,\ B,.......J\]. We have the capacity to accommodate only 5. In how many ways can we arrange them in a line if \[A\] is must and \[G\] and \[H\] must not be included in the team of 5
A) \[^{8}{{P}_{5}}\]
B) \[^{7}{{P}_{5}}\]
C) \[^{7}{{C}_{3}}(4\ !)\]
D) \[^{7}{{C}_{3}}(5\ !)\]
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question_answer22) The number of times the digit 5 will be written when listing the integers from 1 to 1000 is
A) 271
B) 272
C) 300
D) None of these
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question_answer23) The exponent of 3 in \[100\ !\] is
A) 33
B) 44
C) 48
D) 52
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question_answer24) The total number of different combinations of one or more letters which can be made from the letters of the word 'MISSISSIPPI' is
A) 150
B) 148
C) 149
D) None of these
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question_answer25) A person goes in for an examination in which there are four papers with a maximum of \[m\] marks from each paper. The number of ways in which one can get \[2m\] marks is
A) \[^{2m+3}{{C}_{3}}\]
B) \[\frac{1}{3}(m+1)(2{{m}^{2}}+4m+1)\]
C) \[\frac{1}{3}(m+1)(2{{m}^{2}}+4m+3)\]
D) None of these
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question_answer26) There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves proved to exceed by 66 the number of games that the men played with the women. The number of participants is
A) 6
B) 11
C) 13
D) None of these
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question_answer27) A father with 8 children takes them 3 at a time to the Zoological gardens, as often as he can without taking the same 3 children together more than once. The number of times each child will go to the garden is
A) 56
B) 21
C) 112
D) None of these
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question_answer28) A library has \[a\] copies of one book, \[b\] copies of each of two books, \[c\] copies of each of three books and single copies of \[d\] books. The total number of ways in which these books can be distributed is
A) \[\frac{(a+b+c+d)\ !}{a\ !\ b\ !\ c\ !}\]
B) \[\frac{(a+2b+3c+d)\ !}{a\ !\ {{(b\ !)}^{2}}{{(c\ !)}^{3}}}\]
C) \[\frac{(a+2b+3c+d)\ !}{a\ !\ b\ !\ c\ !}\]
D) None of these
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question_answer29) A car will hold 2 in the front seat and 1 in the rear seat. If among 6 persons 2 can drive, then the number of ways in which the car can be filled is
A) 10
B) 20
C) 30
D) None of these
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question_answer30) There are \[(n+1)\] white and \[(n+1)\] black balls each set numbered 1 to \[n+1\]. The number of ways in which the balls can be arranged in a row so that the adjacent balls are of different colours is [EAMCET 1991]
A) \[(2n+2)\ !\]
B) \[(2n+2)\ !\ \times 2\]
C) \[(n+1)\ !\ \times 2\]
D) \[2{{\{(n+1)\ !\}}^{2}}\]
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question_answer31) 12 persons are to be arranged to a round table. If two particular persons among them are not to be side by side, the total number of arrangements is [EAMCET 1994]
A) \[9(10\ !)\]
B) \[2(10\ !)\]
C) \[45(8\ !)\]
D) \[10\ !\]
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question_answer32) How many numbers between 5000 and 10,000 can be formed using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 each digit appearing not more than once in each number [Karnataka CET 1993]
A) \[5{{\times }^{8}}{{P}_{3}}\]
B) \[5{{\times }^{8}}{{C}_{3}}\]
C) \[5\ !\ {{\times }^{8}}{{P}_{3}}\]
D) \[5\ !\ {{\times }^{8}}{{C}_{3}}\]
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question_answer33) If \[x,\ y\] and \[r\] are positive integers, then \[^{x}{{C}_{r}}{{+}^{x}}{{C}_{r1}}^{y}{{C}_{1}}{{+}^{x}}{{C}_{r2}}^{y}{{C}_{2}}+.......{{+}^{y}}{{C}_{r}}=\] [Karnataka CET 1993; RPET 2001]
A) \[\frac{x\ !\ y\ !}{r\ !}\]
B) \[\frac{(x+y)\ !}{r\ !}\]
C) \[^{x+y}{{C}_{r}}\]
D) \[^{xy}{{C}_{r}}\]
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question_answer34) The number of ways in which an arrangement of 4 letters of the word 'PROPORTION' can be made is
A) 700
B) 750
C) 758
D) 800
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question_answer35) The number of different words that can be formed out of the letters of the word 'MORADABAD' taken four at a time is
A) 500
B) 600
C) 620
D) 626
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question_answer36) There are three girls in a class of 10 students. The number of different ways in which they can be seated in a row such that no two of the three girls are together is
A) \[7\ !\ {{\times }^{6}}{{P}_{3}}\]
B) \[7\ !\ {{\times }^{8}}{{P}_{3}}\]
C) \[7\ !\ \times 3\ !\]
D) \[\frac{10\ !}{3\ !\ 7\ !}\]
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question_answer37) For \[2\le r\le n,\left( \begin{matrix} n \\ r \\ \end{matrix} \right)+2\,\left( \begin{align} & \,\,n \\ & r1 \\ \end{align} \right)\]\[+\left( \begin{matrix} n \\ r2 \\ \end{matrix} \right)\] is equal to [IIT Screening 2000; Pb. CET 2000]
A) \[\left( \begin{matrix} n+1 \\ r1 \\ \end{matrix} \right)\]
B) \[2\,\left( \begin{matrix} n+1 \\ r+1 \\ \end{matrix} \right)\]
C) \[2\,\left( \begin{matrix} n+2 \\ r \\ \end{matrix} \right)\]
D) \[\left( \begin{matrix} n+2 \\ r \\ \end{matrix} \right)\]
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question_answer38) The number of positive integral solutions of \[a\,b\,c=30\]is [UPSEAT 2001]
A) 30
B) 27
C) 8
D) None of these
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question_answer39) How many different ninedigit numbers can be formed from the digits of the number 223355888 by rearrangement of the digits so that the odd digits occupy even places [IIT Screening 2000; Karnataka CET 2002]
A) 16
B) 36
C) 60
D) 180
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question_answer40) A dictionary is printed consisting of 7 lettered words only that can be made with a letter of the word CRICKET. If the words are printed at the alphabetical order, as in an ordinary dictionary, then the number of word before the word CRICKET is [Orissa JEE 2003]
A) 530
B) 480
C) 531
D) 481
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