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AIEEE, IITJEE, CAT Problems and solutions in Permutation and Combination

 

“A topic like permutation and combination, which is confusing topic for most of CAT aspirants, can be mastered easily if one uses logic and a little bit of imagination” says Biju Raveendran, CAT trainer.

 

Questions based on Permutation and Combination fall into mainly two categories, and if you master these two categories, you can understand P & C easily.

They are: 1. Similar to Different        2. Different to similar

Similar to Different:

The number of ways of dividing ‘n’ identical (similar) into ‘n’ distinct (different) groups. This type is Similar to Different.

 

A)    Suppose I have 10 identical chocolates which are to be divided into 3 parts where a part can have zero or more chocolates. So let us represent chocolates by blue zeros. The straight red lines are used to divide them into two parts. You can observe that for dividing into 3 parts you need only two lines Suppose yu want to give 1st person 1 chocolate , another person 6 chocolate , then it can be shown as:

Suppose you want to give one person 1 chocolate and another 3 is also can be represented as

For dividing 10 identical chocolates among 3 persons you a can assume that you have 12(10 blue circles and 2 lines) things among which ten are identical of one kind and the remaining two are identical and of another kind.

Hence the number of ways in which you can distribute ten chocolate among 3 people is the same as that of arranging 12 things among which 10 are identical and of one kind while 2 are identical and of another kind. This arrangement can be done is  =  ways.

The above situation is same as finding the number of positive integral solution of a + b + c =10, a, b, c is the number of chocolates given to different persons.

 

Note: Think of dividing 10 chocolates among 4 persons.

The above has no limit (i.e. the group can also be empty)

 

Suppose the group cannot be empty ( It is lower limit question)

In the above example all 3 person should get at least 1, then you have to divide 10 chocolates among 3 persons so that each gets at least one. To solve , in the beginning itself give 1 chocolate to each of them . Now it is a problem of distributing the remaining among them. Now there are 7 chocolates left. Which can be distributed among 3 person such that each gets 0 or more. You can do this in  ways. ( as explained above)

The above situation is same as finding the number of positive natural numbers  of a + b + c =10 , a, b, c is the number of chocolates given to different  persons.

Here are few problems:

(Solutions can be had by e mail)

  1. Rani went to market to buy 18 fruits in all. If there were mangoes, oranges, apples and grapes for sale then in how many ways could Rani buy at least one fruit of each kind?

The Paragraph:

The figure below shows the plan of a town.  The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it . There is also a prohibited region D in the town.

  1. Arum rides his bicycle from house A to Office at B, taking the shortest path. Then the number of possible shortest paths that he can choose is (A) 60  (B) 75  (C) 45  (D) 90  (E) 72
  2. Arum rides his bicycle from house A to club at C , via B taking the shortest path. Then the number of possible paths that he can choose is: (A) 1170   (B) 630   (C) 792   (D) 1200   (E) 936

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