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JEE (Main+Advanced), , Permutation & Combination, , Exercise – I, Section (A) : Fundamental principle of counting, 1., There are 10 buses operating between places A, and B. The number of ways a person can go, from place A to place B and return to place A, if, he returns in a different bus are, (a) 90, (b) 100, (c) 19, (d) 20, 2., The number of numbers from 1000 to 9999, (both inclusive) that do not have all 4 different, digits, is :, (a) 4048, (b) 4464, (c) 4518, (d) 4536, 3., 10 different letters of an alphabet are given., Words with 5 letters are formed from these, given letters, then the number of words which, have at least one letter repeated is :, (a) 69760, (b) 30240, (c) 99748, (d) none, 4., How many three digit even numbers can be, formed using the digits 1,2,3,4,5 (repetition, allowed) ?, (a) 10, (b) 60, (c) 25, (d) 50, 5., The number of three digit odd numbers, that can, be formed by using the digits 1,2,3,4,5,6 when, the repetition is not allowed, is, (a) 60, (b) 108, (c) 36, (d) 30, 6., In a 12 storey house 10 people enter the lift, cabin. It is known that they will leave the lift in, pre-decided groups of 2,3 and 5 people at, different stories. The number of ways they can, do so if the lift does not stop upto the second, storey is (a) 820, (b) 720, (c) 1430, (d) 640, Section (B) : Permutation and combination of distinct, objects gap and string method, Rank of a word, 7., The numbers 1,2,3,4,5 are written on five cards., How many 3 digit numbers can be formed by, placing three cards side by side ?, (a) 60, (b) 30, (c) 12, (d) 10, 8., How many nine digit numbers can be formed, using the digits 2,2,3,3,5,5,8,8,8 so that the odd, digits occupy even positions ?, (a) 7560, (b) 180, , 9., , 10., , 11., , 12., , 13., , 14., , 15., , 16., , 17., , (c) 16, (d) 60, 5 boys & 3 girls are sitting in a row of 8 seats., Number of ways in which they can be seated so, that not all the girls sit side by side, is :, (a) 36000, (b) 9080, (c) 3960, (d) 11600, In how many ways n books can be arranged in a, row so that two specified books are not together, (a) n ! (n  2)! (b) ( n  1)!( n  2), (c) n ! 2( n  1) (d) ( n  2) n !, In how many ways can 5 boys and 5 girls stand, in a row so that boys and girls are alternate ?, (a) 2(5!) 2, , (b) 5! 4!, , (c) 20, , (d) 4 P2 .6 P3, , (c) 9 C2, , (d) 9 P2, , (c) 5! 6!, (d) 6  5!, The sum of the digits in the unit place of all, numbers formed with the help of 3,4,5,6 taken, all at a time is, (a) 18, (b) 432, (c) 108, (d) 144, 8 chairs are numbered from 1 to 8. Two women, & 3 men wish to occupy one chair each. First the, women choose the chairs from amongst the, chairs marked 1 to 4, then the men select the, chairs from among the remaining. The number, of possible arrangements is :, (a) 6 P3, (b) 4 P3, The number of signals that can be made with 3, flags each of different colour by hoisting 1 or 2, or 3 above the other, is :, (a) 3, (b) 7, (c) 15, (d) 16, How many cricket teams of members eleven, each can be formed from 15 persons if captain is, included in every team ?, (a) 364, (b) 1365, (c) 1001, (d) 1000, A bag contains 9 balls marked with digits, 1,2,………………….9. If two balls are drawn, from the bag, then number of ways of getting the, sum of the digits on balls as odd number is (a) 20, (b) 29, In a football championship, 153 matches were, played. Every team played one match with each, other. The number of teams participating in the, championship is -, , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 1

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JEE (Main+Advanced), , 18., , 19., , 20., , 21., , 22., , 23., , Permutation & Combination, , (a) 17, (b) 18, (c) 9, (d) none of these, Out of seven consonants & four vowels, the, number of words of six letters, formed by taking, four consonants & two vowels, is (Assume each, ordered group of letter is a word), (a) 210, (b) 462, (c) 151200, (d) 332640, Out of 16 players of a cricket team, 4 are, bowlers and 2 are wicket keepers. A team of 11, players is to be chosen so as to contain at least 3, bowlers and at least 1 wicketkeeper. The number, of ways in which the team be selected, is, (a) 2400, (b) 2472, (c) 2500, (d) 960, The number of ways in which a mixed double, tennis game can be arranged from amongst 9, married couple if no husband & wife plays in the, same game is :, (a) 756, (b) 3024, (c) 1512, (d) 6048, Six married couple are sitting in a room., Number of ways in which 4 people can be, selected so that there is exactly one married, couple among the four is :, (a) 240, (b) 255, (c) 360, (d) 480, A box contains 2 white balls, 3 black balls & 4, 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 drawn (the balls of the same, colour are different) ., (a) 60, (b) 64, (c) 56, (d) none, Passengers are to travel by a double decked bus, which can accommodate 13 in the upper deck, and 7 in the lower deck. The number of ways, that they can be divided if 5 refuse to sit in the, upper deck and 8 refuse to sit in the lower deck,, is, (a) 25, (b) 21, (c) 18, (d) 15, Words are formed by arranging the letters of the, word “STRANGE” in all possible manner. Let, m be the number of words in which vowels do, not come together and ‘n’ be the number of, words in which vowels come together. Then the, ratio of m : n is (a) 5 : 4, (b) 5 : 2, , (c) 7 : 2, (d) 2 : 5, 25., The sum of all the numbers which can be formed, by using the digits 1,3,5,7 all at a time and, which have no digit repeated, is, (a) 16  4!, (b) 1111 3!, (c) 16 1111 3! (d) 16 1111 4!, Section (C) : Permutation and combination of alike, objects, 26., The number of permutations that can be formed, by arranging all the letters of the word, ‘NINETEEN’ in which no two E’s occur, together, is, (a), , 27., , 28., , 29., , 8!, 3!3!, , (b), , 5!, 3!6 C2, , (c), , 5! 6,  C3, 3!, , (d), , 8! 6,  C3, 5!, , (c), , mn, , Cm, , (d), , 2m  n, , 2m white identical coins and 2n red identical, coins are arranged in a straight line with (m + n), identical coins on each side of a central mark., The number of ways of arranging the identical, coins, so that the arrangements are symmetrical, with respect to the central mark, is, (a) 2 m  2 n C2 m, (b) 2 m 3n Cm, , Cm, , The number of permutations which can be, formed out of the letters of the word “SERIES”, taking three letters together, is :, (a) 120, (b) 60, (c) 42, (d) none, If all the letters of the word “QUEUE” are, arranged in all possible manner as they are in a, dictionary, then the rank of the word QUEUE is, :, (a) 15th, (b) 16th, , (c) 17th, (d) 18th, 30., Sum of all the numbers that can be formed using, all the digits 2,3,3,4,4,4, is :, (a) 22222200, (b) 11111100, (c) 555555500, (d) 20333280, 31., The number of words which can be formed from, 24., the letters of the word “MAXIMUM”, If two, consonants cannot occur together, is, (a) 4!, (b) 3! 4!, (c) 7!, (d) None of these, 32., The number of words that can be formed by, using the letters of the word ‘MATHEMATICS’, that start as well as end with T, is, ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 2

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JEE (Main+Advanced), , Permutation & Combination, , (a) 80720, (b) 90720, (c) 20860, (d) 37528, Section (D) : Division into groups, Selection of one or, more objects, divisors, factorization of numbers, 33., The total number of selections of fruits which, can be made from 3 bananas, 4 apples and 2, oranges is, it is given that fruits of one kind are, identical, (a) 60, (b) 59, (c) 286, (d) 70, 34., Number of ways in which 9 different toys be, distributed among 4 children belonging to, different age groups in such a way that, distribution among the 3 eider children is even, and the youngest one is to receive one toy more,, is :, (a), , 35., , 37., , 38., 39., , 2, , 8, 9!, (c), 3!(2!)3, , 9!, (b), 2, , 36!, (9!) 4, 36!, (c), (9!) 4 .4!, , 41., , 42., , 43., , (d) none, , Number of ways in which a pack of 52 playing, cards be distributed equally among four players, so that each may have the Ace, King, Queen and, Jack of the same suit, is :, (a), , 36., ,  5!, , 40., , (b), , 44., , 36!.4!, (9!) 4, , (d) none, , The number of ways in which the number 27720, can be split into two factors which are coprimes, is :, (a) 15, (b) 16, (c) 25, (d) 49, The number of ways in which the number 94864, can be resolved as a product of two factors is (a) 27, (b) 23, (c) 29, (d) 31, The total number of proper divisors of 38808 is, (a) 60, (b) 59, (c) 286, (d) 70, The number of divisors of a p b1c r d s where, a,b,c,d are primes & p,q,r,s  N, excluding 1 and, the number itself , is :, (a) p q r s, (b) ( p  1)(q  1)(r  1)( s  1)  4, (c) p q r s – 2, , 45., , 46., , (d) ( p  1)(q  1)(r  1)( s  1)  2, How many divisors of 21600 are divisible by 10, but not by 15 ?, (a) 10, (b) 30, (c) 40, (d) none, The sum of the divisors of 25.37.53.7 2 , is, (a) 26.38.54.73, , (b) 26.38.54.73  2.3.5.7, , (a) 4!.4!, , (b), , (a) 2.(4!) 2 (3!) 2, , (b) 2.(3!)3 .4!, , (a) 25, , (b) 26, , (c) 26.38.54.73  1 (d) none of these, Section (E) : Circular permutation, The number of ways in which 6 red roses and 3, white roses (all roses different) can form a, garland so that all the white roses come together,, is, (a) 2170, (b) 2165, (c) 2160, (d) 2155, The number of ways in which 4 boys & 4 girls, can stand in a circle so that each boy and each, girl is one after the other, is :, (a) 3!.4!, (b) 4!.4!, (c) 8!, (d) 7!, 12 guests at a dinner party are to be seated along, a circular table. Supposing that the master and, mistress of the house have fixed seats opposite, to one another and that there are two specified, guests who must always be placed next to one, another. The number of ways in which the, company can be placed, is :, (a) 20.10!, (b) 22.10!, (c) 44.10!, (d) none, The number of ways in which 8 different, flowers can be strung to form a garland so that 4, particular flowers are never separated, is :, , 8!, 4!, , (c) 288, (d) none, Number of ways in which 2 Indians, 3, Americans, 3 Italinans and 4 Frenchmen can be, seated on a circle, If the people of the same, nationality sit together, is :, , (c) 2.(3!) (4!)3 (d) none, Section (F): Multinomial theorem, Distribution of, objects (Method of fictitious partition), 47., Number of positive integral solutions of, x1.x2 .x3  30, is, , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 3

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JEE (Main+Advanced), , 48., , 49., , (c) 27, (d) 28, Number of positive integral solutions of, xyz  21600 is, (a) 1360, (b) 1260, (c) 1460, (d) 1270, If chocolates of a particular brand are all, identical then the number of ways in which we, can choose 6 chocolates out of 8 different brands, available in the market, is :., (a), , 50., , 51., , 52., , 53., , 54., , 13, , C6, , (b), , 13, , (c), , 11, , C4, , (d), , 11, , 58., , 59., , C3, , Number of ways in which 3 persons throw a, normal die to have a total score of 11, is, (a) 27, (b) 25, (c) 29, (d) 18, Section (G) : Geometrical Problems, If 7 points out of 12 are in the same straight line,, then the number of triangles formed is, (a) 185, (b) 466, (c) 462, (d) 286, The number of triangles that can be formed by 5, points in a line and 3 points on a parallel line is, (b) 8 C3 5 C3, , (c) 8 C3 5 C3  1 (d) None of these, , The number of straight lines that can be formed, by joining 20 points no three of which are in the, same straight line except 4 of them which are in, the same line, (a) 183, (b) 186, (c) 197, (d) 185, There are n points in a plane of which ‘p’ points, are collinear. How many lines can be formed, from these points, (a), , C8, , (c) 86, (d) none, The number of ways of selecting 10 balls from, unlimited number of red, black, white and green, balls is, it is given that balls of same colours are, identical, (a) 60, (b) 59, (c) 286, (d) 70, The number of ways of selecting 8 books from a, library which has 10 books each of, Mathematics. Physics, Chemistry and English, if, books of the same subject are alike, is :, (a) 13 C4, (b) 13 C3, , (a) 8 C3, , 55., , Permutation & Combination, , 60., , ( n p ), , C2, , (b) n C2  p C2, , (c) n C2  p C2  1 (d) n C2  p C2  1, , The number of parallelograms that can be, formed from a set of four parallel lines, intersecting another set of three parallel lines is, (a) 6, (b) 18, (c) 12, (d) 9, 61., The greatest possible number of points of, intersection of 8 straight lines and 4 circles is, (a) 32, (b) 64, (c) 76, (d) 104, Section (H) : Exponent of prime number p in n,, Derangement,, 62., Exponent of 3 in 20 ! is (a) 6, (b) 4, (c) 8, (d) 9, 63., Number of zeros at the end of 45! Is (a) 10, (b) 4, (c) 5, (d) 6, 64., A person writes letters to five friends and, addresses on the corresponding envelopes. In, how many ways can the letters be placed in the, envelopes so that all letters are in the wrong, envelopes ?, (a) 20, (b) 40, (c) 44, (d) 109, 65., A person writes letters to five friends and, addresses on the corresponding envelopes. In, how many ways can the letters be placed in the, envelopes so that at least four of them are in the, wrong envelopes ?, (a) 89, (b) 40, (c) 44, (d) 109, , The number of diagonals in a octagon will be, (a) 28, (b) 20, (c) 10, (d) 16, 56., If a polygon has 44 diagonals, then the number, of its sides are, (a) 7, (b) 11, (c) 8, (d) None of these, 57., How many triangles can be formed by joining, four points on a circle, (a) 4, (b) 6, (c) 8, (d) 10, ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 4

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JEE (Main+Advanced), , Permutation & Combination, , Answer Key, 1, a, 11, a, 21, a, 31, a, 41, b, 51, d, 61, d, , 2, b, 12, c, 22, b, 32, b, 42, c, 52, a, 62, c, , 3, a, 13, d, 23, b, 33, b, 43, a, 53, a, 63, a, , 4, d, 14, c, 24, b, 34, c, 44, a, 54, c, 64, c, , 5, a, 15, c, 25, c, 35, b, 45, c, 55, b, 65, a, , 6, b, 16, a, 26, c, 36, b, 46, b, 56, b, , 7, a, 17, b, 27, c, 37, b, 47, c, 57, a, , 8, d, 18, c, 28, c, 38, d, 48, b, 58, d, , 9, a, 19, b, 29, c, 39, d, 49, a, 59, c, , 10, b, 20, c, 30, a, 40, a, 50, c, 60, b, , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 5