Tomato Objective 101-200

  1. The number of ways of distributing 12 identical oranges among children so that every child gets at least one and no child more than 4 is
    (A) 31;
    (B) 52;
    (C) 35;
    (D) 42.
  2. The number of terms in the expansion of \({[(a+3b)^2(a-3b)^2]^2}\) , when simplified, is
    (A) 4;
    (B) 5;
    (C) 6;
    (D) 7.
  3. The number of ways in which 5 persons \({P,Q,R,S}\) and \({T}\) can be seated in a ring so that \(P\) sits between \(Q\) and \(R\) is
    (A) 120;
    (B) 4;
    (C) 24;
    (D) 9.
  4. Four married couples are to be seated in a merry-go-round with 8 identical seats. In how many ways can they be seated so that
    (i) males and females seat alternately, and
    (ii) no husband seats adjacent to his wife?
    (A) 8;
    (B) 12;
    (C) 16;
    (D) 20.
  5. For a regular polygon with \(n\) sides (\(n > 5\)), the number of triangles whose vertices are joining non-adjacent vertices of the polygon is
    (A) \(n(n-4)(n-5)\);
    (B) \((n-3)(n-4)(n-5)/3\);
    (C) \(2(n-3)(n-4)(n-5)\);
    (D) \(n(n-4)(n-5)/6\).
  6. The term that is independent of \(x\) in the expansion of \((\frac {3x^2}{2}\) – \(\frac {1}{3x})^9\) is
    (A) \({\binom{9}{6}}\) \(({\frac{1}{3}})^3\) \(({\frac {3}{2}})^6\);
    (B) \({\binom{9}{5}}\) \(({\frac{3}{2}})^5\) \((-{\frac{1}{3}})^4\);
    (C) \({\binom{9}{3}}\) \(({\frac{1}{6}})^3\)
    (D) \({\binom{9}{4}}\) \(({\frac{3}{2}})^4\) \((-{\frac{1}{3}})^5\);
  7. The value of
    \({\binom{50}{0}} {\binom{50}{1}}+{\binom{50}{1}} {\binom{50}{2}}+…+{\binom{50}{49}} {\binom{50}{50}}\) is
    (A) \({\binom{100}{50}}\);
    (B) \({\binom{100}{51}}\);
    (C) \({\binom{50}{25}}\);
    (D) \({\binom{50}{25}}^2\);
  8. The value of\({\binom{50}{0}}^2+{\binom{50}{1}}^2+{\binom{50}{2}}^2+…+{\binom{50}{49}}^2+{\binom{50}{50}}^2\) is
    (A) \({\binom{100}{50}}\);
    (B) \((50)^{50}\);
    (C) \(2^{100}\);
    (D) \(2^{50}\).
  9. The value of
    \({\binom{100}{0}} {\binom{200}{150}}+{\binom{100}{1}} {\binom{200}{151}}+…+{\binom{100}{50}} {\binom{200}{200}}\) is
    (A) \({\binom{300}{50}}\);
    (B) \({\binom{100}{50}}\) x \({\binom{200}{150}}\);
    (C) \([{\binom{100}{50}}]^2\);
    (D) none of the foregoing numbers.
  10. The number of four-digit numbers strictly greater than 4321 that can be formed from the digits 0,1,2,3,4,5 allowing for repetition of digit is
    (A) 310;
    (B) 360;
    (C) 288;
    (D) 300.
  11. The sum of all the distinct four-digit numbers that can be3 formed using the digits 1,2,3,4 & 5, each digit appearing at most once, is
    (A) 399900;
    (B) 399960;
    (C) 390000;
    (D) 360000.
  12. The number of integers lying between 3000 and 8000 (including 3000 & 8000) which have at least t5wo digits equal is
    (A) 2481;
    (B) 1977;
    (C) 4384;
    (D) 2755.
  13. The greatest integer which, when dividing the integers 13511,13903 and 14593, leaves the same remainder is
    (A) 98;
    (B) 56;
    (C) 2;
    (D) 7.
  14. An integer \(n\) has the property that when divided by 10,9,8,…,2, it leaves remainders 9,8,7,…,1 respectively. A possible value of \(n\) is
    (A) 59;
    (B) 419;
    (C) 1259;
    (D) 2519.
  15. If \(n\) is a positive integer such that \(8n+1\) is a perfect square, then
    (A) \(n\) must be odd;
    (B) \(n\) cannot be a perfect square;
    (C) \(n\) must be a prime number;
    (D) \(2n\) cannot be a perfect square.
  16. For any two integers \(a\) and \(b\) , define \(a{\equiv}b\) if \(a-b\) is divisible by 7. Then (1512+121).(356).(645) \(\equiv\)
    (A) 4;
    (B) 5;
    (C) 3;
    (D) 2.
  17. The coefficient of \(x^2\) in the \binomial equation of \((1+x+x^2)^{10}\) is
    (A) \({\binom{10}{1}}+{\binom{10}{2}}\);
    (B) \({\binom{10}{2}}\);
    (C) \({\binom{10}{1}}\);
    (D) none of the foregoing numbers.
  18. The coefficient of \({x^{17}}\) in the expansion of \({{log}_e(1+x+x^2)}\), where |x| < 1, is
    (A) \({\frac{1}{17}}\)
    (B) \({-\frac{1}{17}}\)
    (C) \({\frac{3}{17}}\)
    (D) none of the foregoing quantities.
  19. Let \({a_1, a_2, . . . , a_{11}}\) be an arbitrary arrangement (i.e. , permutation) of the integers 1,2, . . . , 11. Then the number \({(a_1-1)(a_2-2) . . . (a_{11}-11)}\) is
    (A) necessarily \({\le}\) 0;
    (B) necessarily 0;
    (C) necessarily even;
    (D) not necessarily \({\le}\) 0, 0 or even.
  20. 3 boys of class I, 4 boys of class II and 5 boys of class III sit in a row. The number of ways they can sit, so that boys of that same class sit together is
    (A) \(3!4!5!\);
    (B) \(\frac{(12)!}{3!4!5!}\)
    (C) \((3!)^24!5!\);
    (D) 3 x \(4!5!\)
  21. For each positive integer \(n\) consider the set \(S_n\) defined as follows: \(S_1 = {1}, S_2 = {2,3}, S_3 = {4,5,6}\),…, and, in general, \(S_{n+1}\) consists of \(n+1\) consecutive integers the smallest of which is one more than the largest integer in \(S_n\). Then the sum of all the integers in \(S_{21}\) equals
    (A) 1113;
    (B) 53361;
    (C) 5082;
    (D) 4641.
  22. If the constant term in the expansion of \((\sqrt{x}-\frac{k}{x^2})^{10}\) is 405, then \(k\) is
    (A) \({\pm(3)^\frac{1}{4}}\);
    (B) \({\pm2}\);
    (C) \({\pm(4)^\frac{1}{3}}\);
    (D) \({\pm3}\).
  23. Consider the equation of the form \(x^2+bx+c = 0\). The number of such equations that have real roots and have coefficients \(b\) and \(c\) in the set {1,2,3,4,5,6}, (b may be equal to c), is
    (A) 20;
    (B) 18;
    (C) 17;
    (D) 19.
  24. The number of polynomials of the form \(x^3+ax^2+bx+c\) which are divisible by \(x^2+1\) and where \(a,b\) and \(c\) belong to {1,2,…,10}, is
    (A) 1;
    (B) 10;
    (C) 11;
    (D) 100.
  25. The number of distinct 6-digit numbers between ! and 300000 which are divisible by 4 and are obtained by rearranging the digits of 112233, is
    (A) 12;
    (B) 15;
    (C) 18;
    (D) 90.
  26. The number of odd positive integers smaller than or equal to 10000 which are divisible neither by 3 nor by 5 is
    (A) 3332;
    (B) 2666;
    (C) 2999;
    (D) 3665.
  27. The number of ways one can put three ball numbered 1,2,3 in three boxes labelled a,b,c such that at the most one box is empty is equal to (A) 6;
    (B) 24;
    (C) 42;
    (D) 18.
  28. A bag contains coloured balls of which at least 90% are red. Balls are drawn from the bag one by one and their colour noted. It is found that 49 of the first 50 balls drawn are red. Thereafter 7 out of every 8 balls drawn are red. The number of balls in the bag CAN NOT BE
    (A) 170;
    (B) 210;
    (C) 250;
    (D) 194.
  29. There are N boxes, each containing at most r balls. If the number of boxes containing at least \(i\) balls is \(N_i\) for \(i = 1,2,…,r,\) then the total number of balls contained in these \(N\) boxes
    (A) cannot be determined from the given information;
    (B) is exactly equal to \(N_1+N_2+…+N_r\);
    (C) is strictly larger than \(N_1+N_2+…+N_r\);
    (D) is strictly smaller than \(N_1+N_2+…+N_r\).
  30. For all \(n\), the value of \({\binom{2n}{n}}\) is equal to
    (A) \({\binom{2n}{0}-\binom{2n}{1}+\binom{2n}{2}-\binom{2n}{3}+…+\binom{2n}{2n}}\);
    (B) \({\binom{2n}{0}^2+\binom{2n}{1}^2+\binom{2n}{2}^2+…+\binom{2n}{n}^2}\);
    (C) \({\binom{2n}{0}^2-\binom{2n}{1}^2+\binom{2n}{2}^2-\binom{2n}{3}^2+…+\binom{2n}{2n}^2}\);
    (D) none of the foregoing expressions.
  31. The coefficients of three consecutive terms in the expansion of \((1+x)^n\) are 165, 330 and 462. Then the value of \(n\) is
    (A) 10;
    (B) 12;
    (C) 13;
    (D) 11.
  32. The number of ways in which 4 persons can be divided into two equal groups is
    (A) 3;
    (B) 12;
    (C) 6;
    (D) none of the foregoing numbers.
  33. The number of ways in which 8 persons numbered 1,2,…,8 can be seated in a ring so that 1 always sits between 2 and 3 is
    (A) 240;
    (B) 360;
    (C) 72;
    (D) 120.
  34. There are seven greetings cards, each of a different color, and seven envelopes of the same seven colours. The number of ways in which the cards can be put in the envelopes, so that exactly four of the cards go into the envelopes of the right colours, is
    (A) \({\binom{7}{3}}\);
    (B) \(2{\binom{7}{3}}\);
    (C) \((3!){\binom{4}{3}}\);
    (D) \((3!){\binom{7}{3}\binom{4}{3}}\).
  35. The number of distinct positive integers that can be formed using 0,1,2,4 where each integer is used at the most once is equal to
    (A) 48;
    (B) 84;
    (C) 64;
    (D) 36.
  36. A class contains three girls and four boys. Every Saturday five students go on a picnic, a different group being sent each week. During the picnic, each girl in the group is given a doll by the accompanying teacher. After all possible groups of five have gone once, the total number of dolls the girls have got is
    (A) 27;
    (B) 11;
    (C) 21;
    (D) 45.
  37. From a group o0f seven persons, seven committees are formed. Any two of committees have exactly one member in common. each person is in exactly three committees. Then
    (A) at least one committee must have more than three members;
    (B) each committee must have exactly three members;
    (C) each committee must have more than three members;
    (D) nothing can be said about the sizes of the committee.
  38. Three ladies have each brought a child for admission to school. The head of the school wishes to interview the six people one by one, taking care that no child is interviewed before its mother. In how many different ways can the interviews be arranged?
    (A) 6;
    (B) 36;
    (C) 72;
    (D) 90.
  39. The coefficient of \(x^4\) in the expansion of \((1+x-2x^2)^7\) is
    (A) -81;
    (B) -91;
    (C) +81;
    (D) +91.
  40. The coefficient of \(a^3b^4c^5\) in the expansion of \((bc+ca+ab)^6\) is
    (A) \(\frac{(12)!}{3!4!5!}\);
    (B) \({\binom{6}{3}3!}\);
    (C) 33;
    (D) \({3\binom{6}{3}}\);
  41. The coefficient of \(t^3\) in the expansion of \((\frac {1-t^{6}}{1-t})^3\) is
    (A) 10;
    (B) 12;
    (C) 18;
    (D) 0.
  42. The value of \({\binom{2n}{0}^2-\binom{2n}{1}^2+\binom{2n}{2}^2-…-\binom{2n}{2n-1}^2+\binom{2n}{2n}^2}\) is
    (A) \({\binom{4n}{2n}}\)
    (B) \({\binom{2n}{n}}\)
    (C) 0;
    (D) \((-1)^n{\binom{2n}{n}}\)
  43. There are 14 intermediate stations between Dusi and Visakhapatnam on the South Eastern Railway. A train is to be arranged from Dusi to Visakhapatnam so that it halts at exactly three intermediate stations, no two of which are consecutive. Then the number of ways of doing this is
    (A) \({\binom{14}{3}-\binom{13}{1}\binom{12}{1}+\binom{12}{1}}\);
    (B) \(\frac {10x11x12}{6})\)
    (C) \({\binom{14}{3}-\binom{14}{2}-\binom{14}{1}}\);
    (D) \({\binom{14}{3}-\binom{14}{2}+\binom{14}{1}}\).
  44. The letters of the word “MOTHER” are permuted, and all the permutations so formed are arranged in alphabetical order as in a dictionary. Then the number of permutations which come before the word “MOTHER” is
    (A) 503;
    (B) 93;
    (C) \(\frac{6!}{2}-1\);
    (D) 308.
  45. All the letters of the word PESSIMISTIC are to be arranged so that no two S’s occur together, no two I’s occur together, and S,I do not occur together.
    (A) 2400;
    (B) 5480;
    (C) 48000;
    (D) 50400.
  46. Suppose that \(x\) is an irrational number and \(a,b,c,d\) are rational numbers such that \(\frac{ax+b}{cx+d}\) is rational. Then it follows that
    (A) a=c=0;
    (B) a=c & b=d;
    (C) a+b=c+d;
    (D) ad=bc.
  47. Let \(p,q\)  and $latex s$ be integers such that \(p^2=sq^2\). Then it follows that
    (A) \(p\) is an even number;
    (B) if \(s\) divides \(p\), then \(s\) is a perfect square;
    (C) \(s\) divides \(p\);
    (D) \(q^2\) divides \(p\).
  48. The number of pairs of positive integers \((x,y)\) where \(x\) and \(y\) are prime numbers and \(x^2-y^2=1\) is
    (A) 0;
    (B) 1;
    (C) 2;
    (D) 8.
  49. A point P with coordinates \((x,y)\) is said to be good if both x and y are positive integers. The number of good points on the curve \(xy = 27027\) is
    (A) 8;
    (B) 16;
    (C) 32;
    (D) 64.
  50. Let \(p\) be an odd prime number. Then the number of positive integers \(k\) with \(1<k<p\), for which \(k^2\) leaves a remainder of 1 when divided by \(p\), is
    (A) 2;
    (B) 1;
    (C) \(p-1\);
    (D) \(\frac{p-1}{2}\).
  51. Let \(n=51!+1\). Then the number of primes among \(n+1, n+2,…,n+50\) is
    (A) 2;
    (B) 1;
    (C) 2;
    (D) more than 2.
  52. If three prime numbers, all greater than 3, are in A.P., then their common difference
    (A) must be divisible by 2 but not necessarily by 3;
    (B) must be divisible by 3 but not necessarily by 2;
    (C) must be divisible by both 2 and 3.
    (D) need not be divisible by any of 2 and 3.
  53. Let \(N\) be a positive integer not equal to 1. Then note that none of the numbers 2,3,…,N is a divisor of (N!-1). From this, we can conclude that
    (A) (N!-1) is a prime number;
    (B) at least one of the numbers \(N+1,N+2,…,N!-2\) is a divisor of (N!-1).
    (C) the smallest number between N and N! which is a divisor of (N!-1), is a prime number;
    (D) none of the foregoing statements is necessarily correct.
  54. The number 1000!=1.2.3….1000 ends exactly with
    (A) 249 zeros;
    (B) 250 zeros;
    (C) 240 zeros;
    (D) 200 zeros.
  55. Let A denote the set of all prime numbers, B the set of all prime numbers and the number 4, and C the set of all prime numbers and their squares. Let D be the set of positive integers k, for which \(\frac{k-1}{k}\) is not an integer. Then
    (A) D=A;
    (B) D=B;
    (C) D=C;
    (D) B \(\subset\) D \(\subset\) C.
  56. Let \(n\) be any integer. Then \(n(n+1)(2n+1)\)
    (A) is a perfect square;
    (B) is an odd number;
    (C) is an integral multiple of 6;
    (D) does not necessarily have any of the foregoing properties.
  57. The numbers \(12n+1\) and \(30n+2\) are relatively prime for
    (A) any positive integer \(n\).
    (B) infinitely many, but not all, integers \(n\).
    (C) for finitely many integers \(n\).
    (D) none of the above.
  58. The expression \(1+\frac{1}{2}\binom{n}{1}+\frac{1}{3}\binom{n}{2}+\frac{1}{n+1}\binom{n}{n}\) equals
    (A) \(\frac{2^{n+1}-1}{n+1}\)
    (B) \(2\frac{(2^{n}-1)}{n+1}\)
    (C) \(\frac{2^n-1}{n+1}\)
    (D) \(2\frac{(2^{n+1}-1)}{n+1}\)
  59. The value of
    (A) \(\frac{2^{31}}{30}\);
    (B) \(\frac{2^{30}}{31}\);
    (C) \(\frac{2^{31}-1}{30}\);
    (D) \(\frac{2^{30}-1}{31}\);
  60. The value of
    where M-k>100, k>100
    and \(\binom{m}{k}=\frac{m!}{n!(m-n)!}\) equals
    (A) \(\frac{k}{M}\);
    (B) \(\frac{M}{k}\);
    (C) \(\frac{k}{M^2}\);
    (D) \(\frac{M}{k^2}\).
  61. The remainder obtained when \(1!+2!+…+95!\) is divided by 15 is
    (A) 14;
    (B) 3;
    (C) 1;
    (D) none of the foregoing numbers.
  62. Let \(x_1,x_2,…,x_{50}\) be fifty integers such that the sum of any six of them is24. Then
    (A) the largest of \(x_i\) equals 6;
    (B) the smallest of \(x_i\) equals 3;
    (C) \(x_{16}=x_{34}\);
    (D) none of the foregoing statements is correct.
  63. Let \(x_1,x_2,…,x_{50}\) be fifty non zero numbers such that \(x_i+x_{i+1}=k\) for all i., 1 \({le}\) i \({le}\) 49.
    If \(x_{14}=a, x_{27}=b\), then \(x_{20}+x_{37}\) equals
    (A) 2(a+b)-k;
    (B) k+a;
    (C) k+b;
    (D) none of the foregoing expressions.
  64. Let S be the set of all numbers of the form \(4^n-3n-1\), where \(n=1,2,3,\)…
    Let T be the set of all numbers of the form \(9(n-1)\), where \(n=1,2,3,\)…
    Only one of the following statements is correct. Which one is it?
    (A) Each number in S is also in T;
    (B) Each number in T is also in S;
    (C) Every number in S is in T and every number in T is in S;
    (D) There are numbers in S which are not in T and there are numbers in T which are not in S.
  65. The number of four-digit numbers greater than 5000 that can be formed out of the digits 3,4,5,6 and 7, no digit being repeated, is
    (A) 52;
    (B) 61;
    (C) 72;
    (D) 80.
  66. The number of positive integers of 5 digits such that each digit is 1,2 or 3, and all three of the digits appear at least once, is
    (A) 243;
    (B) 150;
    (C) 147;
    (D) 193.
  67. In a class tournament, each of the 5 players plays against every other player. No game results in a draw and the winner of each game gets one point and the loser gets zero. Then which one of the following sequences cannot represent the scores of the five players?
    (A) 3,3,2,1,1;
    (B) 3,2,2,2,1;
    (C) 2,2,2,2,2;
    (D) 4,4,1,1,0.
  68. Ten persons numbered 1,2,…,10 play a chess tournament, each player playing against every other player exactly one game. Assume that each game results in a win for one of the players.
    Let \(w_1,w_2,…,w_{10}\) be the number of games won by players 1,2,…,10 respectively. Also, let \(l_1,l_2,…,l_{10}\) be the number of games lost by the players 1,2,…,10 respectively.
    (A) \(w^2_1+w^2_2+…+w^2_{10}= 81-(l^2_1+l^2_2+…+l^2_{10})\);
    (B) \(w^2_1+w^2_2+…+w^2_{10}= 81+(l^2_1+l^2_2+…+l^2_{10})\);
    (C) \(w^2_1+w^2_2+…+w^2_{10}=l^2_1+l^2_2+…+l^2_{10}\);
    (D) none of the foregoing equalities is necessarily true.
  69. A game consisting of 10 rounds is played among three players A,B and C as follows: Two players play i9n each round and the loser is replaced by the third player in the next round. If the only rounds when A played against B are the first, fourth and the tenth rounds, the number of games won by C is
    (A) 5;
    (B) 6;
    (C) 7;
    (D) cannot be determined by the above information.
  70. An n x n chess board is a square of sides n units which has been sub-divided into \(n^2\) unit squares by equally-spaced straight lines parallel to the sides. The total number of squares of all sizes on an n x n chess board is
    (A) \(\frac{n(n+1)}{2}\);
    (B) \(1^2+2^2+…+n^2\);
    (C) 2 x 1+3 x 2+4 x 3+…+n x (n-1)
    (D) given by none of the foregoing expressions.
  71. Given any five points in the square
    \(I^2=\) {\((x,y): 0\le\) \(x\le\) \(1,0\le\) \(y\le\) \(1\)}, only one of the following statements is true. Which one is it?
    (A) The five points lie on a circle.
    (B) At least one square can be formed using four of the five points.
    (C) At least three of the five points are collinear.
    (D) There are at least two points such that the distance between them does not exceed \(\frac{1}{\sqrt2}\).
  72. The quantities l,c,h and m are measured in the units mentioned against each:
    l: centimetre;
    c: centimetre per second;
    h: ergs x second:
    \(mc^2\): ergs. of the expressions \({\alpha}\) = \(\mathbf{\frac{ch}{ml}^\frac{1}{2}}\)
  73. The number of distinct rearrangements of the letters of the word “MULTIPLE” that can be made preserving the order in which the vowels (U,I,E) occur and not counting the original arrangement is
    (A) 6719;
    (B) 3359;
    (C) 6720;
    (D) 3214.
  74. The number of terms in the expansion of \((x+y+z+w)^{10}\) is
    (A) \({\binom{10}{4}}\);
    (B) \({\binom{13}{3}}\);
    (C) \({\binom{14}{4}}\);
    (D) \(11^4\).
  75. The number of ways in which three non-negative integers \(n_1,n_2,n_3\) can be chosen such that \(n_1+n_2+n_3=10\) is
    (A) 66;
    (B) 55;
    (C) \(10^3\);
    (D) \(\frac{(10)!}{3!2!1!}\)
  76. In an examination, the score in each of the four languages – Bengali, Hindi, Urdu and Telegu- can be integers between 0 and 10. Then the number of ways in which a student can secure a total score of 21 is
    (A) 880;
    (B) 760;
    (C) 450;
    (D) 1360.
  77. The number of ordered pairs (x,y) of positive integers such that (x+y)=90 and their greatest common divisor is 6 equals
    (A) 15;
    (B) 14;
    (C) 8;
    (D) 10.
  78. How many pairs of positive integers (m,n) are there satisfying \(m^3-n^3=21\)?
    (A) exactly one;
    (B) none;
    (C) exactly three;
    (D) infinitely many.
  79. The number of ways in which three distinct numbers in A.P can be selected from 1,2,…,24 is
    (A) 144;
    (B) 276;
    (C) 572;
    (D) 132.
  80. The number of ways you can invite 3 of your friends on 5 consecutive days, exactly one friend a day, such that no friend is invited on more than two days is
    (A) 90;
    (B) 60;
    (C) 30;
    (D) 10.
  81. Consider three boxes, each containing 10 balls labelled 1,2,…,10. suppose one ball is drawn from each of the boxes. Denote by \(n_i\), the label of the ball drawn from the i-th box, i=1,2,3.
    Then the number of ways in which the balls can be chosen such that \(n_1<n_2<n_3\), is
    (A) 120;
    (B) 130;
    (C) 150;
    (D) 160.
  82. The number of sequences of length five with 0 and 1 as terms which contain at least two consecutive zeros is,
    (A) \(4.2^3\);
    (B) \(\binom{5}{2}\);
    (C) 20;
    (D) 19.
  83. There are 7 identical white balls and 3 identical black balls. The number of distinguishable arrangements in a row of all the balls, so that no two black balls are adjacent, is
    (A) 120;
    (B) 89(8!);
    (C) 56;
    (D) 42x\(5^4\).
  84. In a multiple-choice test there are 6 questions. 4 alternatives answers are given for each question by choosing one answer for each question, then the number of ways to get exactly 4 correct answers is
    (A) \(4^6-4^2\);
    (B) 135;
    (C) 9;
    (D) 120.
  85. In a multiple-choice test there are 8 questions. Each question has 4 alternatives, of which only one is correct. If a candidate answers all the questions by choosing one alternative for each, the number of ways of doing it so that exactly 4 answers are correct is
    (A) 70;
    (B) 2835;
    (C) 5670;
    (D) none of the foregoing numbers.
  86. Among the 8! permutations of digits 1,2,3…,8 consider those arrangements which have the following property: if you take any five consecutive positions, the product of the digits in those positions is divisible by 5. The number of such arrangements is
    (A) 7!;
    (B) 2.7!;
    (C) 8.7!
    (D) 4.\({\binom{7}{4}}\)5!3!4
  87. A closet has 5 pairs of shoes. The number of ways in which 4 shoes can be chosen from it so that there will be no complete pair, is
    (A) 80;
    (B) 160;
    (C) 200;
    (D) none of the foregoing numbers.
  88. The number of ways in which 4 distinct balls can be put into 4 boxes labelled a,b,c,d so that exactly one box remain empty is
    (A) 232;
    (B) 196;
    (C) 192;
    (D) 144.
  89. The number of permutations of the letters a,b,c,d such that b does not follow a, and c does not follow b, and d does not follow c, is
    (A) 12;
    (B) 11;
    (C) 14;
    (D) 13.
  90. The number of ways of seating three gentlemen and three ladies in a row, such that each gentleman is adjacent to at least one lady, is
    (A) 360;
    (B) 72;
    (C) 720;
    (D) none of the foregoing numbers.
  91. the number of maps \(f\) from the set {1,2,3} into the set {1,2,3,4,5} such that
    \(f(i)\) \({\le}\) \(f(j)\), whenever \(i<j\)
    (A) 30;
    (B) 35;
    (C) 50;
    (D) 60.
  92. For each integer \({\mathbf{i}}\), 1 \({\le}\) i \({\le}\) 100;
    \({\epsilon}_i\) be either +1 or -1. Assume that \({\epsilon}_1\) = +1 and \({\epsilon}_{100}\) = -1. Say that a sign change occurs at i \({\ge}\) 2 if \({\epsilon}_i\), \({\epsilon}_{i-1}\) are of opposite sign. then the total number of sign changes
    (A) is odd;
    (B) is even;
    (C) is at most 50;
    (D) can have 49 distinct values.
  93. Let \(S={1,2,…,n}\). The number of possible pairs of the form (A,B) with A \({\subseteq}\) B for subsets A and B of \(S\) is
    (A) \(2^n\);
    (B) \(3^n\);
    (C) \({\sum_{k=0}^{n}\binom{n}{k}\binom{n}{n-k}}\)
    (D) n!.
  94. There are 4 pairs of shoes of different sizes. Each of the 8 shoes can be coloured with one of the four colours: Black, Brown, White & Red. In how many ways can one colour shoes so that in at least three pairs, the left and the right shoes do not have the same colour?
    (A) \(12^4\);
    (B) 28x\(12^3\);
    (C) 16x\(12^3\);
    (D) 4x\(12^3\).
  95. Let \(S={1,2,…,100}\). the number of non empty subsets A of S such that the product of elements in A is even is
    (A) \(2^{50}(2^{50}-1)\);
    (B) \(2^{100}-1\);
    (C) \(2^{50}-1\);
    (D) none of these numbers.
  96. The number of functions f from {1,2,…,20} onto {1,2,…,20} such that f(k) is a multiple of 3 whenever k is a multiple of 4 is
    (A) 5!.6!.9!;
    (B) \(5^6.15!\);
    (C) \(6^5.14!\);
    (D) 15!.6!.
  97. Let \(X={a_1,a_2,…,a_7}\) be a set of seven elements and \(Y={b_1,b_2,b_3}\) a set of three elements. The number of functions f from X to Y such that
    (i) f is onto and
    (ii) there are exactly three elements x in X such that \(f(x)=b_1\), is
    (A) 490;
    (B) 558;
    (C) 560;
    (D) 1680.
  98. Consider the quadratic equation of the form \(x^2+bx+c=0\). The number of such equations that have real roots and coefficient b and c from the set {1,2,3,4,5} (b and c may be equal) is
    (A) 18;
    (B) 15;
    (C) 12;
    (D) none of the foregoing quantities.
  99. Let \(A_1,A_2,A_3\) be three points on a straight line. Let \(B_1,B_2,B_3,B_4,B_5\) be five points on a straight line parallel to the first one. Each of the three points on the first line joined by a straight line to each of the five points on the second line. further, no three or more of these joining lines meet at a point except possible at the A’s or the B’s. Then the number of points of intersections of the joining lines lying between the two given straight lines is
    (A) 30;
    (B) 25;
    (C) 35;
    (D) 20.
  100. There are 11 points on a plane with % lying on one straight line and another 5 lying on a second straight line which is parallel to the first line. The remaining point is not collinear with any two of the previous 10 points. The number of the triangle that can be formed with vertices chosen from these 11 points is
    (A) 85;
    (B) 105;
    (C) 125;
    (D) 145.


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