IITPrep

Code : MPCP1

Mathematics

Elementary Combinatorics & Probability


  1. There are 10 long nails and 20 rings. All the 20 rings are to be placed on the 10 nails. Each nail is long enough to accommodate all the 20 rings if so desired. We take no account of the absolute position of a ring, but we do take into account of its position relative to other rings on the same nail. All the rings and nails are distinguishable. Find the number of ways in which the rings can be arranged on the nails in the desired fashion.
  2. Show that the number of ways of selecting n objects out of 3n objects, of which n are alike and the other 2n are different is 22n-1 + (2n-1)! /[ n! (n-1)! ]
  3. Given n pairs of gloves, Find the number of ways in which each of n persons can pick up a right and a left handed glove which do not form a pair.
  4. There are 3 copies of 1 book, 2 copies of another and 1 copy of a 3rd. They have to be distributed among 20 persons in such a way that (i) none receives more than 1 book (ii) none receives 2 copies of the same book but may get 2 or 3 different books. Find the number of possible distributions in each case.
  5. In how many ways can we construct non-congruent triangles whose sides are integers > n and < 2n? How many of them are isosceles (count equilateral also) and how many are equilateral?
  6. The sides of a triangle are a, b, c where they are all integers and a £ b £ c. If c is given, Show that the number of isosceles or equilateral triangles is (3c-2)/2 or (3c-1)/2 according as c is even or odd.
  7. There are 10 couples to be divided into 5 groups, 4 in each.
    1. In how many ways can they be split up so that there are 2 men and 2 women in each group ?
    2. In how many of these ways will a given man be in the same group as his wife ?
    3. And in how many of these ways will 2 given men be with their wives?
  8. In how many distinct ways can the face of a cube be painted, each face with a different color, given 6 different colours? Two modes of painting are considered same if one can be carried into other by rigid motions of the cube.
  9. Missiles are fired at a target. The probability of each missile hitting the target is P. The hits are independent of one another. Each missile hitting the target brings it down with probability r. The missiles are fired until the target is brought down or the missile reserve is exhausted. The reserve consists of n (> 2) missiles. Defining the event A as at least 1 missile will remain in reserve. Find P(A).
  10. Four roads lead away from a jail. A prisoner is escaping from a jail and selects a road at random. If road I is selected the probability of escaping is 1/8, for road II its 1/6, road III ¼ and for road IV 9/10. What is the probability that the prisoner escapes?
  11. A and B are independent mathematicians liable to error. The chances of their solving a problem are 1/8 and 1/12. It is 1000 to 1 against their making the same mistake. If they get the same result, find the probability that the result is correct.
  12. A three men Jury has two members each of whom independently has probability P of making the correct decision and third flips a coin for each decision. The decision of majority becomes final. A one man Jury has probability P of making the right decision. Which Jury has the better probability of making the correct decision?
  13. A game of craps is played as follows. The player throws 2 dice and the player wins if the total for his 1st throw is 7 or 11, loses at once if its 2, 3, 12. Any other throw is called his point. If the first throw is a point, the player throws the dice repeatedly until he either wins by throwing his point again or loses by throwing a 7. Find the players’ chance of winning.
  14. An urn contains n balls each of different color of which one is white. 2 independent observers each with probability 0.1 of telling the truth assert that a ball drawn at random is white. Find the probability that the ball is in fact white.
  15. All possible 8 digit numbers are formed using 1,3,4,5,6,7,8,9 without repetition. Find the probability of a number selected at random to be divisible by 275.
  16. On a straight line AB of length a + b, two segments PQ, P’Q’ of lengths a, b respectively are measured at random. If c is less than a or b, Prove that the chance that the common part of PQ, P’Q’ is less than c is c2/ab.
  17. How many different numbers which are smaller than 2.108 and are divisible by 3, can be written by means of the digits 0, 1, 2?
  18. How many 4-digit numbers can be formed whose decimal notation contains not more than two different digits?