Page 260 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 12: Probability and Expected Value

Notes

Tasks A committee of 8 teachers is to be formed out of 6 science, 8 arts teachers
and a physical instructor. In how many ways the committee can be formed if
1. Any teacher can be included in the committee.
2. There should be 3 science and 4 arts teachers on the committee such that (i) any
science teacher and any arts teacher can be included, (ii) one particular science teacher
must be on the committee, (iii) three particular arts teachers must not be on the
committee?

12.3.4 Ordered Partitions

1. Ordered Partitions (distinguishable objects)
(a) The total number of ways of putting n distinct objects into r compartments which
are marked as 1, 2, ...... r is equal to r .
n
Since first object can be put in any of the r compartments in r ways, second can be put
in any of the r compartments in r ways and so on.
(b) The number of ways in which n objects can be put into r compartments such that the
first compartment contains n objects, second contains n objects and so on the rth
1 2
compartment contains n objects, where n + n + ...... + n = n, is given by
r 1 2 r
n!
.
n !n ! ...... n !
1
r
2
To illustrate this, let r = 3. Then n objects in the first compartment can be put in
1
n
C ways. Out of the remaining n - n objects, n objects can be put in the second
n 1 1 2
n- n
compartment in 1 C ways. Finally the remaining n - n - n = n objects can be put
n 2 1 2 3
in the third compartment in one way. Thus, the required number of ways is
! n
n n 1 n
C C .
1 n 2 n n ! ! !
n
n
1 2 3
2. Ordered Partitions (identical objects)
(a) The total number of ways of putting n identical objects into r compartments marked
n+r- 1
as 1, 2, ...... r, is C r- 1 , where each compartment may have none or any number
of objects.
We can think of n objects being placed in a row and partitioned by the (r - 1) vertical
lines into r compartments. This is equivalent to permutations of (n + r - 1) objects out
of which n are of one type and (r - 1) of another type. The required number of
n r 1 ! n r 1
permutations are , which is equal to n r 1 C or C r 1 .
n! r 1 ! n
(b) The total number of ways of putting n identical objects into r compartments is
n r r 1 n 1
C r 1 or C r 1 , where each compartment must have at least one object.

In order that each compartment must have at least one object, we first put one object
in each of the r compartments. Then the remaining (n - r) objects can be placed as in
(a) above.

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