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Division and Distribution of Objects - Permutation and Combination

Posted on - 23-02-2017

JEE Math PC

IIT JEE

Division and distribution of objects

(with fixed number of objects in each group)

Into groups of unequal size (different number of objects in each group):

(a) Number of ways in which n distinct objects can be divided into r unequal groups containing a1 objects in the first group, a2 objects in the second group and so on

= =

Here a1 + a2 + a3 + … +ar = n.

(b) Number of ways in which n distinct objects can be distributed among r persons such that first person gets a1 objects, 2nd person gets a2 objects…,rth person gets ar objects = .

Explanation:

Let us divide the task into two parts . In the first part, we divide the objects into groups. In the second part, these r groups can be assigned to r persons in r! ways. .

Into groups of equal size (each group containing same number of objects):

(a) Number of ways in which m´n distinct objects can be divided equally into n groups (unmarked) =

(b) 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)! =

Example.1

If out of 50 players, 5 teams of 10 players each have to be formed this becomes a question on grouping and thus required number of ways to form such teams is

Derangement

Let S = {1, 2, 3, …. ,n}, then a function f from S to S known as derangement if f is a bijective function and f(i) > i for any i ∈ S.

In other words rearrangement of objects such that no one goes to its original place is called derangement

If 'n' things are arranged in a row, the number of ways in which they can be deranged so that none of them occupies its original place is

= n!and it is denoted by D(n)

Note:

The above result can be obtained by using inclusion exclusion principle. For this you can refer to problem –13 on page 29.

Example.2

Suppose 4 letters are taken out of 4 different envelopes. In how many ways, can they be reinserted in the envelopes so that no letter goes in to its original envelope ?.

Solution

Using the formula for the number of derangements that are possible out of 4 letters in 4 envelopes, we get the number of ways as :

 
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