Vol XXXIV No 12 55 63 40 31 December 2016 Corporate Office: Plot 99, Sector 44 Institutional Area, Gurgaon -122 003 (HR), Tel : 0124-6601200 e-mail : info@mtg.in website : www.mtg.in 71 24 Regd Office: 406, Taj Apartment, Near Safdarjung Hospital, Ring Road, New Delhi - 110029 Managing Editor : Mahabir Singh Editor : Anil Ahlawat 70 contents 38 Class XI 67 Concept Boosters 82 24 Brain @ Work 31 Ace Your Way (Series 8) Subscribe online at 38 MPP-6 Class XII 40 Concept Boosters 55 Ace Your Way (Series 8) 63 Challenging Problems 70 Maths Musing Problem Set - 168 71 Mock Test Paper - WB JEE 78 Olympiad Corner 82 You Ask We Answer 85 Maths Musing Solutions www.mtg.in Individual Subscription Rates Mathematics Today Chemistry Today Physics For You Biology Today yr yrs yrs 330 330 330 330 600 600 600 600 775 775 775 775 Combined Subscription Rates 67 MPP-6 Competition Edge 78 yr PCM PCB PCMB 900 900 1000 yrs yrs 1500 1500 1800 1900 1900 2300 Send D.D/M.O in favour of MTG Learning Media (P) Ltd Payments should be made directly to : MTG Learning Media (P) Ltd, Plot 99, Sector 44 Institutional Area, Gurgaon - 122 003, Haryana We have not appointed any subscription agent Owned, Printed and Published by Mahabir Singh from 406, Taj Apartment, New Delhi - 29 and printed by Personal Graphics and Advertisers (P) Ltd., Okhla Industrial Area, Phase-II, New Delhi Readers are advised to make appropriate thorough enquiries before acting upon any advertisements published in this magazine Focus/Infocus features are marketing incentives MTG does not vouch or subscribe to the claims and representations made by advertisers All disputes are subject to Delhi jurisdiction only Editor : Anil Ahlawat Copyright© MTG Learning Media (P) Ltd All rights reserved Reproduction in any form is prohibited mathematics today | DECEMBER‘16 Permutations and Combinations This column is aimed at Class XI students so that they can prepare for competitive exams such as JEE Main/Advanced, etc and be also in command of what is being covered in their school as part of NCERT syllabus The problems here are a happy blend of the straight and the twisted, the simple and the difficult and the easy and the challenging z z The Factorial : Let n be a positive integer Then, the continued product of first n natural numbers is called factorial n, to be denoted by n ! or n Also, we define ! = When n is negative or a fraction, n ! is not defined Thus, n ! = n (n – 1) (n – 2) 3·2·1 Exponent of prime p in n!: Let p be a prime number and n is a positive integer (i.e natural number) If EP(n) denote the exponent of the prime p in the positive integer n, then exponent of prime p in n! is denoted by Ep(n!) and defined by NUMbER OF PERMU TATIONS wIThOU T REPETITION z Arranging n objects, taken r at a time equivalent to filling r places from n things n n n = + + + k p p p where k is the largest positive integer satisfying pk ≤ n < pk + z r�places : Number of choices : n (n 1)(n 2)(n 3) mathematics today | DECEMBER‘16 n (r 1) The number of ways of arranging = The number of ways of filling r places = n(n – 1)(n – 2) (n – r + 1) = Ep(n!) = Ep (1 (n – 1)n) PERMUTATIONS The ways of arranging a smaller or an equal number of persons or objects at a time from a given group of persons or objects with due regard being paid to the order of arrangement are called the (different) permutations The number of all Permutations of n things taking r at a time is denoted by npr npr is always a natural number n! n and pr = (n − r)! Three different things a, b and c are given, then different arrangements which can be made by taking two things from three given things are ab, ac, bc, ba, ca, cb Therefore the number of permutations will be r n(n − 1)(n − 2) (n − r + 1)((n − r)!) n! = =nPr (n − r)! (n − r)! The number of arrangements of n different objects taken all at a time = nPn = n! n! • n P0 = = 1; nPr = n n−1Pr −1 n! = or (−r)! = ∞ (r ∈ N ) • ! = 1; (−r)! N U M b E R O F P E R M U TAT I O N S w I T h REPETITION z The number of permutations (arrangements) of n different objects, taken r at a time, when each object may occur once, twice, thrice, upto r times in any arrangement = The number of ways of filling r places where each place can be filled by any one of n objects r�places : r Number of choices : n n n n n z The number of permutations = The number of ways of filling r places = (n)r The number of arrangements that can be formed using n objects out of which p are identical (and of one kind) q are identical (and of another kind), r are identical (and of another kind) and the rest are n! distinct is p !q !r ! CONdITIONAl PERMUTATIONS z Number of permutations of n dissimilar things taken r at a time when p particular things always occur = rPp· n – pPr – p z Number of permutations of n dissimilar things taken r at a time when p particular things never occur = n – pPr z The total number of permutations of n different things taken not more than r at a time, when each thing may be repeated any number of times, is n(nr − 1) n −1 z Number of permutations of n different things, taken all at a time, when m specified things always come together is m! × (n – m + 1)! z Number of permutations of n different things, taken all at a time, when m specified things never come together is n! – m! × (n – m + 1)! Let there be n objects, of which m objects are alike z of one kind, and the remaining (n – m) objects are alike of another kind Then, the total number of mutually distinguishable permutations that can be n! formed from these objects is (m !) × (n − m)! The above theorem can be extended further i.e., if there are n objects, of which p1 are alike of one kind; p2 are alike of 2nd kind; p3are alike of 3rd kind; ; prare alike of rth kind such that p1 + p2 + pr = n; then the number of permutations of these n objects is n! ( p1 !) × ( p2 !) × × ( pr !) CIRCUlAR PERMUTATIONS In circular permutations, what really matters is the position of an object relative to the others Thus, in circular permutations, we fix the position of the one of the objects and then arrange the other objects in all possible ways 10 mathematics today | DECEMBER‘16 There are two types of circular permutations : z The circular permutations in which clockwise and the anticlockwise arrangements give rise to different permutations, e.g Seating arrangements of persons round a table z The circular permutations in which clockwise and the anticlockwise arrangements give rise to same permutations, e.g arranging some beads to form a necklace Theorems on circular permutations z The number of circular permutations of n different objects when clockwise and anticlockwise are taken as different is (n – 1)! z The number of ways in which n different object when clockwise and anticlockwise are not not different is (n − 1)! z Number of circular permutations of n different things, taken r at a time, when clockwise and anticlockwise orders are taken as different is n z Pr r Number of circular permutations of n different things, taken r at a time, when clockwise and anticlockwise orders are not different is n Pr 2r COMbINATIONS Each of the different groups or selections which can be formed by taking some or all of a number of objects, irrespective of their arrangements, is called combination The number of all combinations of n things, taken z n r at a time is denoted by C (n, r) or nCr or r nC r is always a natural number and n! n Cr = r !(n − r)! difference between permutation and combination : z In a combination only selection is made whereas in a permutation not only a selection is made but also an arrangement in a definite order is considered z E ach combinat ion cor resp onds to many permutations For example, the six permutations ABC, ACB, BCA, BAC, CBA and CAB correspond to the same combination ABC NUMbER OF C OMbINATIONS wIThOU T REPETITION The number of combinations (selections or groups) that can be formed from n different objects taken r(0 ≤ r ≤ n) at a time is n! n Also nCr = nCn−r Cr = r !(n − r)! Let the total number of selections (or groups) = x Each group contains r objects, which can be arranged in r ! ways Hence the number of arrangements of r objects =x × (r!) But the number of arrangements = nPr ⇒ x(r !) = nPr ⇒ x = n Pr n! ⇒x= =nCr r! r !(n − r)! NUMbER OF COMbINATIONS wITh REPETITION ANd All POSSIblE SElECTIONS z The number of combinations of n distinct objects taken r at a time when any object may be repeated any number of times = Coefficient of xr in (1 + x + x2 + + xr)n = Coefficient of xr in (1 – x)–n = n + r – 1Cr The total number of ways in which it is possible to z form groups by taking some or all of n things at a time is nC1 + nC2 + + nCn = 2n – z The total number of ways in which it is possible to make groups by taking some or all out of n = (n1 + n2 + ) things, when n1 are alike of one kind, n2 are alike of second kind, and so on is {n1 + 1)(n2 + 1) } – z The number of selections of r objects out of n identical objects is Total number of selections of zero or more objects z from n identical objects is n + z The number of selections taking at least one out of a1 + a2 + a3 + + an + k objects, where a1 are alike (of one kind), a2 are alike (of second kind) and so on an are alike (of nth kind) and k are distinct = [(a1 + 1)(a2 + 1)(a3 + 1) (an + 1)]2k – CONdITIONAl COMbINATIONS z The number of ways in which r objects can be selected from n different objects if k particular objects are • Always included =n – kCr – k • Never included = n – k Cr The number of combinations of n objects, of which z p are identical, taken r at a time is 12 mathematics today | DECEMBER‘16 n – pC r + n – pCr – + n – pCr – + + n – p C0, if r ≤ p and n – pC + n – pC n – pC n – pC r r–1+ r – + + r – p, if r > p dIvISION INTO gROUPS Case I The number of ways in which n different things z can be arranged into r different groups when order of the groups are considered is n + r –1Pn or n ! n – 1C r – 1according as blank group are or are not admissible z The number of ways in which n different things can be distributed into r different groups when order of groups are not considered is rn – rC1(r – 1)n + rC2(r – 2)n – + (–1)r – 1rCr – or coefficient of xn in n !(ex – 1)r Here blank groups are not allowed z Number of ways in which m × n different objects can be distributed equally among n persons (or numbered groups) = (number of ways of dividing into groups) × (number of groups) ! (mn)!n ! (mn)! = = (m !)nn ! (m !)n Case II z The number of ways in which (m + n) different things can be divided into two groups which contain m and n (m + n)! things respectively is, m+nCm.nCn = , m ≠ n m !n ! Corollary: If m = n, then the groups are equal size Division of these groups can be given by two types • I f order of group is not important : The number of ways in which 2n different things can be divided equally into two groups is (2n)! !(n !)2 • I f order of group is important : The number of ways in which 2n different things can be divided equally into two distinct groups is (2n)! 2n ! × 2! = 2 !(n !) (n !)2 z The number of ways in which (m + n + p) different things can be divided into three groups which contain m, n and p things respectively is (m + n + p)! m +n + p Cm.n+ pCn pC p = , m ≠ n ≠ p m !n ! p ! Corollary : If m = n = p, then the groups are equal size Division of these groups can be given by two types • I f order of group is not important : The number of ways in which 3p different things can be divided equally into three groups is (3 p)! !( p !)3 • I f order of group is important : The number of ways in which 3p different things can be divided equally into three distinct groups is (3 p)! (3 p)! 3! = 3 !( p !) ( p !)3 • If order of group is not important : The number of ways in which mn different things can be divided equally into m groups is mn ! (n !)m m ! • I f order of group is important : The number of ways in which mn different things can be divided equally into m distinct groups is (mn)! (mn)! × m! = m (n !) m ! (n !)m dEARRANgEMENT Any change in the given order of the things is called a de-arrangement If n things form an arrangement in a row, the number of ways in which they can be de-arranged so that no one of them occupies its original place is 1 1 n ! 1 − + − + + (−1)n n! 1! ! ! SOME IMPORTANT RESUlTS z Number of total different straight lines formed by joining the n points on a plane of which m (< n) are collinear is nC2 – mC2 + Number of total triangles formed by joining the n z points on a plane of which m (< n) are collinear is nC3 – mC3 z Number of diagonals in a polygon of n sides is nC – n z If m parallel lines in a plane are intersected by a family of other n parallel lines Then total number of parallelograms so formed is mn(m − 1)(n − 1) m C2 ×nC2 i.e., z Given n points on the circumference of a circle, then z • Number of straight lines = nC2 • Number of triangles = nC3 • Number of quadrilaterals = nC4 If n straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent Then the number of part into which these lines divide the plane is = + Sn MUlTINOMIAl ThEOREM Let x1, x2, ., xm be integers Then number of solutions to the equation x1 + x2 + + xm = n (i) Subject to the condition a1 ≤ x1 ≤ b1, a2 ≤ x2 ≤ b2, am ≤ xm ≤ bm (ii) is equal to the coefficient of xn in (x a1 + x a1+1 + + x b1)(x a2 + x a2 +1 + + x b2 ) .(x am + x am+1 + + x bm ) (iii) This is because the number of ways, in which sum of m integers in (i) equals n, is the same as the number of times xn comes in (iii) z Use of solution of linear equation and coefficient of a power in expansions to find the number of ways of distribution : (i) The number of integral solutions of x1 + x2 + x3 + + xr = n where x1 ≥ x2 ≥ 0, xr ≥ is the same as the number of ways to distribute n identical things among r persons This is also equal to the coefficient of xn in the expansion of (x0 + x1 + x2 + x3 + )r r z = coefficient of x n in 1− x n + r −1 Cr −1 = coefficient of xn in (1 – x)–r = The number of integral solutions of x1 + x2 + x3 + + xr = n where x1 ≥ 1, x2 ≥ 1, xr ≥ is same as the number of ways to distribute n identical things among r persons each getting at least This also equal to the coefficient of xn in the expansion of (x1 + x2 + x3 + )r r = coefficient of xn in x 1 − x = coefficient of xn in xr(1 – x)–r NUMbER OF dIvISORS Let N = p1a1 p2a2 p3a3 pkak , where p1, p2 , p3, pk are different primes and a1, a2 , a3 , , ak are natural numbers then : z The total number of divisors of N including and N is = (a1 + 1)(a2 + 1)(a3 + 1) (ak + 1) mathematics today | DECEMBER‘16 13 z The sum of these divisors is z a a = ( p10 + p11 + p12 + + p1 )( p20 + p12 + p22 + + p2 ) a ( pk0 + p1k + pk2 + + pk k ) z The number of ways in which N can be resolved as a product of two factors is 1 (a1 + 1)(a2 + 1) (a k + 1), If N is not a perfect square [(a + 1)(a + 1) (a + 1) + 1], If N is a perfect square k 2 z The number of ways in which a composite number N can be resolved into two factors which are relatively prime (or co-prime) to each other is equal to 2n – where n is the number of different factors in N SOME MORE TEChNIQUES nC = nC = 1, nC = n z n nC + nC n + 1C z r r– = r nC = nC ⇔ x = y or x + y = n z x y n ⋅ n – 1Cr – = (n – r + 1) nCr – z z If n is even then the greatest value of nCr is nCn/2 z If n is odd then the greatest value of nCr is or z z z z z z z n n C n−1 14 z z come together = z C n+1 n Cr = n−1Cr −1 r Number of selections of zero or more things out of n different things is, nC0 + nC1 + nC2 + + nCn = 2n The number of ways of answering all of n questions when each question has an alternative is 2n n C + n C + n C + = n C + n C + n C + = 2n – 2n + 1C + 2n + 1C + 2n + 1C + 2n + 1C = 22n n nC + n + 1C + n + 2C + n + 3C + + 2n – 1C n n n n n = 2nCn + Number of combinations of n dissimilar things taken n! n !(n − n)! = = 1, (0 ! = 1) 0! gap method : Suppose males A, B, C, D, E are arranged in a row as × A × B × C × D × E × There will be six gaps between these five Four in between and two at either end Now if three females P, Q,R are to be arranged so that no two are together we shall use gap method i.e., arrange them in between these gaps Hence the answer will be 6P3 mathematics today | DECEMBER‘16 m !n ! type come together = all at a time nCn = z n z Together : Suppose we have to arrange persons in a row which can be done in ! = 120 ways But if two particular persons are to be together always, then we tie these two particular persons with a string Thus we have – + (1 corresponding to these two together) = +1 = units, which can be arranged in 4! ways Now we loosen the string and these two particular can be arranged in 2! ways Thus total arrangements = 24 × = 48 Never together = Total – Together = 120 – 48 = 72 The number of ways in which n (one type of different) things and n (another type of different) things can be arranged in a row alternatively is ⋅ n! ⋅ n! The number of ways in which m things of one type and n things of another type can be arranged in the form of a garland so that all the second type of things and no two things of second (m − 1)! mPn If we are given n different digits (a, a2, a3 an) then sum of the digits in the unit place of all numbers formed without repetition is (n – 1)! (a1 + a2 + a3 + + an) Sum of the total numbers in this case can be obtained by applying the formula (n – 1)!(a1 + a2 + a3 + an) (1111 n times) problems Single Correct Answer Type How many numbers can be formed from the digits 1, 2, 3, when the repetition is not allowed (a) (c) 4P 4P + 4P + 4P (b) (d) 4P 4P + P2 + P3 + P4 How many even numbers of different digits can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, (repetition is not allowed) (a) 224 (c) 324 (a) (b) 280 (d) None of these The value of nPr is equal to n – 1P r + r n – Pr – (b) n n−1Pr +n−1Pr −1 (c) n(n−1Pr +n−1Pr −1) (d) n −1 Pr −1 +n−1 Pr Find the total number of digit numbers which have all the digits different (a) × 9! (b) 9! (c) 10! (d) None of these The sum of the digits in the unit place of all numbers formed with the help of 3, 4, 5, (without repetition) taken all at a time is (a) 18 (b) 432 (c) 108 (d) 144 The figures 4, 5, 6, 7, are written in every possible order The number of numbers greater than 56000 is (a) 72 (b) 96 (c) 90 (d) 98 The sum of all digit numbers that can be formed by using the digits 2, 4, 6, (repetition of digits is not allowed) is (a) 133320 (b) 533280 (c) 53328 (d) None of these The number of words which can be formed from the letters of the word MAXIMUM, if two consonants cannot occur together, is (a) 4! (b) 3! × 4! (c) 7! (d) None of these 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! 10 Numbers greater than 1000 but not greater than 4000 which can be formed with the digits 0, 1, 2, 3, (repetition of digits is allowed), are (a) 350 (b) 375 (c) 450 (d) 576 11 The number of numbers that can be formed with the help of the digits 1, 2, 3, 4, 3, 2, so that odd digits always occupy odd places, is (a) 24 (b) 18 (c) 12 (d) 30 12 In a circus there are ten cages for accommodating ten animals Out of these four cages are so small that five out of 10 animals cannot enter into them In how many ways will it be possible to accommodate ten animals in these ten cages (a) 66400 (b) 86400 (c) 96400 (d) None of these 13 All possible four digit numbers are formed using the digits 0, 1, 2, so that no number has repeated digits The number of even numbers among them is (a) (b) 18 (c) 10 (d) None of these 14 The number of ways in which ten candidates A1 · A2, , A10 can be ranked such that A1 is always above A10 is (a) 5! (b) 2(5!) (c) 10! (d) (10 !) 15 In how many ways can boys and girls stand in a row so that no two girls may be together (a) (5!)2 (b) 5! × 4! (c) 5! × 6! (d) × 5! 16 How many numbers greater than hundred and divisible by can be made from the digits 3, 4, 5, 6, if no digit is repeated (a) (b) 12 (c) 24 (d) 30 17 The number of digit even numbers that can be formed using 0, 1, 2, 3, 4, 5, without repetition is (a) 120 (b) 300 (c) 420 (d) 20 18 Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, (repitition is allowed) are (a) 216 (b) 375 (c) 400 (d) 720 19 The number of words that can be formed out of the letters of the word ARTICLE so that the vowels occupy even places is (a) 36 (b) 574 (c) 144 (d) 754 20 How many numbers lying between 999 and 10000 can be formed with the help of the digits 0,2,3,6,7,8 when the digits are not to be repeated (a) 100 (b) 200 (c) 300 (d) 400 21 20 persons are invited for a party In how many different ways can they and the host be seated at a circular table, if the two particular persons are to be seated on either side of the host (a) 20! (b) · 18! (c) 18! (d) None of these 22 The number of ways in which beads of different colours form a necklace is (a) 12 (b) 24 (c) 120 (d) 60 23 In how many ways men and women can be seated around a round table such that no two women can sit together (a) (7!)2 (b) 7! × 6! (c) (6!)2 (d) 7! 24 If a =mC2 , then a C2 is equal to (a) m + 1C4 (b) m – 1C4 m + (c) · C4 (d) · m + 1C4 mathematics today | DECEMBER‘16 15 (a) − (b) 25 −1 x 10 If f (x) = 1− x 3 − x (c) for (d) − 25 x < −1 for −1 ≤ x ≤ for < x < x ≥2 for Then lim f (x) and lim f (x) equal to x →2+ x →2− (a) 1, –1 (c) 0, a b 11 lim 1 + + x x x →∞ (b) –1, –1 (d) 0, –1 2x (a) a = 1, b = (c) a = 1, b ∈ R = e , then (b) a = 2, b = (d) a = 1, b = 12 If f : R → R be a differentiable function, such that f (x + 2y) = f (x) + f (2y) + 4xy " x, y ∈ R, then (a) f ′(1) = f ′(0) + (b) f ′(1) = f ′(0) – (c) f ′(0) = f ′(1) + (d) f ′(0) = f ′(1) – p sin{cos x} ,x≠ p x− 13 If f (x) = p x= 1, where {⋅} represents the fractional part, then (a) f(x) is continuous at x = p/2 (b) lim f (x) exists, but f is not continuous x →p/2 at x = p/2 (c) lim f (x) does not exist x →p/2 (d) lim f (x) = −1 x → p /2 x− x [ ], x (x − 3) is continuous at x = 4, then (a) a = 0, b = (b) a = 1/2, b = (c) a = –1/2, b = 72 mathematics today | DECEMBER‘16 (d) f (x) is continuous everywhere for any real a and b a |sin (1+ |sin x |) x| , − p < x < 15 If the function f (x) = b, x =0 tan 2x p tan 3x , 0 (1 + sin px)t = lim = −1 when sin px < t →∞ when sin px = 1+ (1 + sin px)t 1− \ Range of f(x) = {–1, 0, 1} (b) : Graph of y = f (x) is symmetrical about the line x = 0, if f (–x) = f (x) i.e if f (0 – x) = f (0 + x) \ Graph of y = f (x) is symmetrical about x = 1, if f (1 – x) = f (1 + x) (d) : Given that f (x) = 2xn + a And f (2) = 26, f (4) = 138 \ f (2) = · 2n + a = 26 (1) 74 mathematics today | DECEMBER‘16 And, f (4) = · 4n + a = 138 (2) From (1) and (2) we get, · 4n – · 2n = 112 ⇒ (2n)2 – 2n – 56 = ⇒ (2n – 8)(2n + 7) = ⇒ n = \ f (x) = 2x3 + a Now from (1) we get, · 23 + a = 26 ⇒ a = 10 \ f (x) = 2x3 + 10 Thus, f (3) = · 33 + 10 = 64 (a) : In order that f (x) be defined log3cos(sinx) ≥ ⇒ cos(sinx) ≥ ⇒ cos(sinx) = (Q –1 ≤ cosq ≤ 1) ⇒ sinx = ⇒ x = np, n ∈ I n n−1 (a) : Let f (x) = a0 x + a1x + + an 1 1 Since, f (x) ⋅ f = f (x) + f we get, x x a a (a0 x n + a1x n−1 + + an ) ⋅ 0n + n1−1 + + an x x a a = (a0 x n + a1x n−1 + + an ) + 0n + n1−1 + + an x x On comparison of co-efficient of like powers of x we get a0 = ±1 and an = and a1 = a2 = = an – = ⇒ f (x) = xn + 1, f (10) = 1001 = 10n + ⇒ n = ⇒ f (x) = xn + 1, f (20) = 203 + = 8001 (a) : The function will be defined f (x) − f (x) when |f (x)| – f (x) > ⇒ |f (x)| > f (x) ≥ f (x) < ⇒ logex + logxe < (Q f (x) = logex + logxe) (log e x)2 + 1 ⇒ log e x +