
question_answer1) The value of \[{{2}^{n}}\,\{1.3.5.....(2n3)\,(2n1)\}\] is
A) \[\frac{(2n)\,!}{n\,!}\] done
clear
B) \[\frac{(2n)\,!}{{{2}^{n}}}\] done
clear
C) \[{{2}^{ni}}\] done
clear
D) None of these done
clear
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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 done
clear
B) 100 done
clear
C) 200 done
clear
D) None of these done
clear
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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 done
clear
B) 810 done
clear
C) 2346 done
clear
D) None of these done
clear
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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 done
clear
B) 2400 done
clear
C) 14400 done
clear
D) None of these done
clear
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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 done
clear
B) 7 done
clear
C) 9 done
clear
D) None of these done
clear
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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 done
clear
B) 41 done
clear
C) 51 done
clear
D) None of these done
clear
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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 done
clear
B) 30240 done
clear
C) 99748 done
clear
D) None of these done
clear
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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) \[\frac{52\ !}{{{(17\ !)}^{3}}}\] done
clear
B) \[52\ !\] done
clear
C) \[\frac{52\ !}{17\ !}\] done
clear
D) None of these done
clear
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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 done
clear
B) 900 done
clear
C) 1260 done
clear
D) 1620 done
clear
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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 done
clear
B) 45 done
clear
C) 46 done
clear
D) None of these done
clear
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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) \[\frac{m\ !\ (m+1)\ !}{(mn+1)\ !}\] done
clear
B) \[\frac{m\ !\ (m1)\ !}{(mn+1)\ !}\] done
clear
C) \[\frac{(m1)\ !\ (m+1)\ !}{(mn+1)\ !}\] done
clear
D) None of these done
clear
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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 done
clear
B) 240 done
clear
C) 600 done
clear
D) 3125 done
clear
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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 done
clear
B) 11 done
clear
C) 12 done
clear
D) 13 done
clear
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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 done
clear
B) 300 done
clear
C) 271 done
clear
D) 302 done
clear
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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) \[\frac{10\ !}{6}\] done
clear
B) \[3\ !\ 7\ !\] done
clear
C) \[^{10}{{P}_{3}}\ .\ 7\ !\] done
clear
D) None of these done
clear
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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) \[^{21}{{C}_{7}}\] done
clear
B) \[^{30}{{C}_{16}}\] done
clear
C) \[^{21}{{C}_{16}}\] done
clear
D) None of these done
clear
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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 done
clear
B) 16630 done
clear
C) 1663200 done
clear
D) None of these done
clear
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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 done
clear
B) 100 done
clear
C) 150 done
clear
D) 200 done
clear
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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 done
clear
B) 246 done
clear
C) 252 done
clear
D) None of these done
clear
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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 done
clear
B) 300 done
clear
C) 240 done
clear
D) 120 done
clear
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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) \[^{8}{{P}_{5}}\] done
clear
B) \[^{7}{{P}_{5}}\] done
clear
C) \[^{7}{{C}_{3}}(4\ !)\] done
clear
D) \[^{7}{{C}_{3}}(5\ !)\] done
clear
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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 done
clear
B) 272 done
clear
C) 300 done
clear
D) None of these done
clear
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question_answer23) The exponent of 3 in \[100\ !\] is
A) 33 done
clear
B) 44 done
clear
C) 48 done
clear
D) 52 done
clear
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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 done
clear
B) 148 done
clear
C) 149 done
clear
D) None of these done
clear
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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) \[^{2m+3}{{C}_{3}}\] done
clear
B) \[\frac{1}{3}(m+1)(2{{m}^{2}}+4m+1)\] done
clear
C) \[\frac{1}{3}(m+1)(2{{m}^{2}}+4m+3)\] done
clear
D) None of these done
clear
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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 done
clear
B) 11 done
clear
C) 13 done
clear
D) None of these done
clear
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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 done
clear
B) 21 done
clear
C) 112 done
clear
D) None of these done
clear
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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) \[\frac{(a+b+c+d)\ !}{a\ !\ b\ !\ c\ !}\] done
clear
B) \[\frac{(a+2b+3c+d)\ !}{a\ !\ {{(b\ !)}^{2}}{{(c\ !)}^{3}}}\] done
clear
C) \[\frac{(a+2b+3c+d)\ !}{a\ !\ b\ !\ c\ !}\] done
clear
D) None of these done
clear
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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 done
clear
B) 20 done
clear
C) 30 done
clear
D) None of these done
clear
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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) \[(2n+2)\ !\] done
clear
B) \[(2n+2)\ !\ \times 2\] done
clear
C) \[(n+1)\ !\ \times 2\] done
clear
D) \[2{{\{(n+1)\ !\}}^{2}}\] done
clear
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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) \[9(10\ !)\] done
clear
B) \[2(10\ !)\] done
clear
C) \[45(8\ !)\] done
clear
D) \[10\ !\] done
clear
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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) \[5{{\times }^{8}}{{P}_{3}}\] done
clear
B) \[5{{\times }^{8}}{{C}_{3}}\] done
clear
C) \[5\ !\ {{\times }^{8}}{{P}_{3}}\] done
clear
D) \[5\ !\ {{\times }^{8}}{{C}_{3}}\] done
clear
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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) \[\frac{x\ !\ y\ !}{r\ !}\] done
clear
B) \[\frac{(x+y)\ !}{r\ !}\] done
clear
C) \[^{x+y}{{C}_{r}}\] done
clear
D) \[^{xy}{{C}_{r}}\] done
clear
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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 done
clear
B) 750 done
clear
C) 758 done
clear
D) 800 done
clear
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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 done
clear
B) 600 done
clear
C) 620 done
clear
D) 626 done
clear
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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) \[7\ !\ {{\times }^{6}}{{P}_{3}}\] done
clear
B) \[7\ !\ {{\times }^{8}}{{P}_{3}}\] done
clear
C) \[7\ !\ \times 3\ !\] done
clear
D) \[\frac{10\ !}{3\ !\ 7\ !}\] done
clear
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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) \[\left( \begin{matrix} n+1 \\ r1 \\ \end{matrix} \right)\] done
clear
B) \[2\,\left( \begin{matrix} n+1 \\ r+1 \\ \end{matrix} \right)\] done
clear
C) \[2\,\left( \begin{matrix} n+2 \\ r \\ \end{matrix} \right)\] done
clear
D) \[\left( \begin{matrix} n+2 \\ r \\ \end{matrix} \right)\] done
clear
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question_answer38) The number of positive integral solutions of \[a\,b\,c=30\]is [UPSEAT 2001]
A) 30 done
clear
B) 27 done
clear
C) 8 done
clear
D) None of these done
clear
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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 done
clear
B) 36 done
clear
C) 60 done
clear
D) 180 done
clear
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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 done
clear
B) 480 done
clear
C) 531 done
clear
D) 481 done
clear
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