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 :
