permutation and combinations problems

7/27/2019 permutation and combinations problems

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Permutation and combination (50 Problems)

Questions 1-15:

1) Find the number of ways of distribution of 12 identical balls in three identicalboxes.

2) a) How many words can be made using the letters AEROLUX all at a time?

b) How many of these begin and end with a consonant?

c) How many of these have the vowels and consonants occupying thesame relativepositions?

d) Find the number of words which can be formed out of the letters a, b, c, d, e, f taken 3together, each word containing one vowel at least.

3) a) In how many ways can the letters in the word MISSISSIPPI be arranged?

b)In how many ways can they be arranged if any 2 Ss must be separated?

4) Suppose crews of 43 people are used to steer a ship. There are 3 among themwho cansteer and we need only 1 person to steer. Out of the remaining 40people. 8 can row only on oneside and 3 people can only row on the other side.How many ways can the crew be arranged inso that there are 20 rowers on each side of the ship?

5) Find the permutations of different things taken not more than at a time wheneach thingmay be repeated any number of times.

6) a) Find the largest integer for which is divisible by .

b) Prove that the equation has just one solution in the set ofnatural numbers.

c) For denote the greatest integer less than or equal to , then find the value of

d) Prove that is divisible by .

7) a) Suppose we have 10 points in a plane forming a decagon.


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(i) How many triangles can be formed by joining them?

(ii) How many diagonals are there in the decagon?

b) A train going from Delhi to Jaipur stops at 7 intermediate stations. Five persons enter the trainduring the journey with five different tickets of the same class. How many different sets of ticketsthey could had?

8) There were two women participating in a chess tournament every participant played twogames with every other participant. The number of games that men played between themselvesproved to exceed by 66, compared to the number of games the men played with women. Howmany participants were there? How many games were played?

9) a) Suppose n different games are to be given to n children. In how many ways canthis be done so that exactly one child gets no game.

b) Two packs of 52 playing cards are shuffled together. Find the number of ways in which aman can be dealt 20 cards so that he does not get two cards of the same suit and samedenomination.

c) In how many ways can 10 different prizes be given to 5 students, so that one boy getexactly 4 prizes and the rest of the students can get any number of prizes.

10) a) Mr. A has children by his first wife and Ms. B has children by her first husband.They marry and have children of their own. The whole family has 10 children. Assuming that twochildren of the same parents do not fight, find the maximum number of fights that can take placeamong children.

b) If all permutations of the letters of the word RAKSHIT are arranged as in a dictionary, then findthe rank the word RAKSHIT.

11) In how many ways can a team of 3 chemistry teachers and 4 mathematics teachers can beformed from 8 chemistry teachers and 10 mathematics teachers such that a particular chemistryteacher refuses to be in the team if a particular mathematics is in the team.

12) Find the number of ways in which an examiner can assign 30 marks to 8 questions, givingnot less than 2 marks to any question.

13) a) In how many ways can 2 squares be selected from an chessboard so that

(i) they are not in the same row or the same column ;

(ii) they have one side in common ;

(iii) they are in the same diagonal.


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b) In how many ways we can place two queens (one black and one white) on anchessboard so that their fire lines remains separate.

14) a) Show that a selection of 10 balls can be made from an unlimited number of red, white,blue and green balls in 286 different ways and that 84 of these contain balls of all four colours.

b) Find the number of ways of selecting 5 coins from coins, three each of Rs. 1, Rs 2 and Rs. 5.

15) a). If n objects are arranged in a row, then find the number of ways of selecting three of theseobjects so that no two of them are next to each other.

b) If distinct things are arranged in a circle, show that the number of ways selecting three of

these things so that no two of them are next to each other is .

Questions: 16-30

1. Find the number of positive integers from 1 to 106 (both inclusive) which are neitherperfect squares, nor perfect cubes, nor perfect fourth powers.

2. In a row of 20 seats, in how many ways can three blocks of consecutive seats with five

seats in each block be selected.

3. A mathematics paper has 12 questions divided into 3 sections . each having 4questions. In how many ways can you answer 5 questions selecting at least one question fromeach part.

4. We have 4 balls of different colours and 4 boxes with colours the same as those of the

balls. In how many ways can the balls be arranged in the boxes so that no ball goes into a box ofits own colour.

5. A person wants to hold as many different parties as he can out of 24 friends, each partyconsisting of the same number. How many should he invite at a time? In how many of thesewould the same man be found?

6. A box contains two white balls, three black balls and four red balls. In how many wayscan three balls be drawn from the box if at least one black ball is to be included in the draw.

7. In how man wa s can we lace red balls and white balls in boxes so that each


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box contains at least one ball of each colour.

8. a)A zoo has 25 zebras, 14 giraffes, 16 lions and 2 tigers. In how many ways can a touristvisit these animals so that he must see at least one tiger.

b) Suppose a lunch buffet has 4 sections having 10,9,8,7 different items of cuisine,

respectively. In how many ways, can an invitee to the lunch pick the food items if he is obliged topick at least 4,2,3,1 items from each of the sections respectively.

9. In how many ways can the letters in the English alphabet be arranged, so that there are

7 letters between the letters and .

10. In how many ways can a selection of 5 letters be made from and.

11. There are flowers of one kind and 3 of another kind. Show that the number of

different garlands made of using all these flowers is .

12. a) From a given number of books, there are three sets of identical books onphysics, chemistry and mathematics. The remaining n are distinct books on other subjects. Find

the number of ways of choosing out of books.

b) Show that the number of ways in which things of one sort. of another sort and of a

third sort can be divided between two persons, giving things to each, is .

c) Amongst objects, out of them are identical. Find the number ways select

objects out of these .

d) If out of letters there are As, Bs and Cs, show that the number of ways of

selecting letters out of these is the same as the number of ways of selecting letters out of

them. If , show that the number of ways selecting letters is given by

and that the maximum number of such selections is

or according as is even or odd.

13. a) Three integers are selected from the integers 1, 2, ..., 1000. In how many ways canthese integers be selected such that their sum is divisible by 4.

b) An examination consists of four papers. Each paper has a maximum of m marks. Show that the


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number of ways in which in which a student can get marks in the examination is

.

14. a) How many integral solutions are there to the system of equations

and when .

b) The number of positive integral solutions of the inequality is

15. a) There are m points on one straight line and points on another straight line .

None of them being How many triangles can be formed with these points as vertices ? How

many can be formed if point is also included?

b) Suppose a city has parallel roads running East-West and n parallel roads running North-South. How many rectangles are formed with their sides along these roads? If the distancebetween every consecutive pair of parallel roads is the same, how many shortest possible routesare there to go from one corner of the city to its diagonally opposite comer?

Questions: 31-45

1(a) in How many shortest ways can we reach from the point (0,0,0) to point (3,7,11) in space

where the movement is possible only along the -axis, -axis and -axis or parallel to themand change of axes is permitted only at integral points. (An integral point is one which has its co-ordinate as integers)

(b) In how many ways 21 identical toys can be distributed among 5 children such that each getsat least one toy and no two children get equal number of toys.

2(a) Find the number of integral solutions of , where and .

(b) Find the number of positive integral solutions of .

(c). Find the number of positive, unequal integral solutions of the equation .

3. In a super market there are 8 varieties of articles and each article can be bought in anymanner. In how many ways can one buy 4 articles if

i. there is no restriction

ii. atleast 2 of one variety are to be bought.

iii. atleast 2 of one variety and two of other variety are to be bought.


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4(a) 10 identical/ordinary dice are rolled simultaneously, how many different outcomes arepossible?

(b) 2 distinct, ordinary dice are rolled simultaneously. In have many different ways sum of face

values of dice can be equal to ,

5. Find the number of whole numbers formed on the screen of a calculator which can read upsidedown (i.e. can be recognized as numbers with unique (correct) digits). It is given that greatestnumber that can be formed on the screen of the calculator is 99999999.

6(a) In a big box A, another small boxes are placed. Now in each of these small boxes either n

further smaller boxes are placed or it is left empty. If the total number of filled boxes is equal to ,then find the number of empty boxes.

(b) In how many different ways can the letters of the word BRIJESSH be placed in the 8 boxes ofthe given fig. so that no row remains empty.

7(a) Prove that the totals number of ways in which n' distinct places can be filled by taking any

number things from different things is , where [.] denotes the greatest integer function .

(b)Consider the set of numbers . Two numbers and are selected from this set

with replacement. In how many ways numbers can be selected so that is divisible by 5.

8(a) If we drawn straight lines in the plane consisting of parallel lines in one direction,

parallel lines in the different direction, and parallel lines in another direction such

and no three lines meets in a point. Calculate the total number of points ofintersection.

(b) (i) How many triangles oriented the same way as can be seen in a grid like the oneshown in the figure? The grid consists of rows.

(ii) Express the number of triangles which can be seen the other way up as a sumof binomial coefficients. Also obtain the value of that sum.

9(a) Show that the number of ways in which three numbers in arithmetical progression can be

selected from is or , according as is odd or even.

(b)(i) If and are positive integers such that and , where is a fixed positive

inte er, show that the reatest value of is or


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according as a is even or odd.

(ii) If a straight rods of indefinite length are placed so as to form the greatest possible number of

squares, prove that this number is or , according as a iseven or odd.

10(a) The sides of a triangle are and , where and and . If 'c is given, prove

that number of different triangles is or , according as is even or odd. Also,

show that the number of isosceles of equilateral triangles is or according asis even or odd.

(b) Each side of a triangle is an integral number in cm, no side exceeding cm. Prove that the

number of different triangles which can be so formed is or

, according as c is even or odd. Also show that the number of isosceles or

equilateral triangles is or according as c is even or odd.

11. In how many ways two distinct numbers , and can be selected from the set

so that is a multiple of 5.

12. Find the number of 5-digit numbers, the sum of whose digits leaves a remainder of 2, whendivided by 4.

13. There are seats in the first row of a theatre, of which are to be occupied. Find the numberof ways of arranging persons so that

(i) no two persons sit side by side;

(ii) each person has exactly one neighbor; and

(iii) out of any two seats located symmetrically about the middle of the row at least one is empty.

14. There are n' switches in a gallery numbered from 1 to . A person operates the switches inthe following manner,

(i) he operates each ( starting from one) in succession.

(ii) or, he operates the switches alternatively not necessarily starting from 1


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e.g. or

(iii) or, he operates any combination of (a) and (b) e.g. .

(iv) In any case he operates the switch. Using mathematical induction, prove that the total

number of different ways to operate the switches is .

15. Find the number of all possible distinct triangles having side length as integer and perimeterequal to

.

46.

21. A train going from Delhi to Bombay stops at nine intermediate station. Six persons enter thetrain during the journey with six different tickets of the same class. How many different sets oftickets they have had?

47.

34. In a box there arc 50 tickets, numbered 1, 2, 3,... 50. 5 tickets are taken at a time from the box

and arranged in the order . In how many cases will be equal to 20?

48.

41. A parallelogram is cut by 2 sets of lines parallel to its sides. Show that the total number of

parallelograms thus formed is .

49.

36. An eight - oared boat is to be manned by a crew chosen from 11 men, of whom 3 can steerbut cannot row, and the rest can row but cannot steer. In how many ways can the crew bearranged, if two of the men can only row on the bow side ?

Crew of an ei ht -oared boat consists of 4 rowers on each side and one steerer.


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50. Find number of possible arrangements of 5 married couples, seated alternately man andwomanround a table, 5 wives being in assigned positions and 5 husbands so placed that a man doesnot sitnext to his wife.

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