Page 86 - DMTH404_STATISTICS

Statistics

Notes (c) 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?
Solution.

(a) 2 balls can be selected from 8 balls in 8 C = 8! = 28 ways.
2
2!6!
n n
(b) Since C = C , therefore, the number of groups of 7 persons out of 12 is also equal to
r n- r
the number of groups of 5 persons out of 12. Hence, the required number of groups is
12!
12 .
C = = 792
7
7!5!
Alternative Method. We may regard 7 persons of one type and remaining 5 persons of
another type. The required number of groups are equal to the number of permutations of
12 persons where 7 are alike of one type and 5 are alike of another type.

(c) 1. 8 teachers can be selected out of 15 in 15 C = 15! = 6,435 ways.
8
8!7!
6
2. (i) 3 science teachers can be selected out of 6 teachers in C ways and 4 arts
3
8
teachers can be selected out of 8 in C ways and the physical instructor can be
4
6
1
selected in C way. Therefore, the required number of ways = C  8 C  1 C =
3
1
1
4
20 70 1 = 1400.
5
(ii) 2 additional science teachers can be selected in C ways. The number of
2
selections of other teachers is same as in (i) above. Thus, the required number
5
of ways = C  8 C  1 C = 10 70 1 = 700.
2
4
1
6
(iii) 3 science teachers can be selected in C ways and 4 arts teachers out of remaining
3
5
5 arts teachers can be selected in C ways.
4
6
 The required number of ways = C  5 C = 20 5 = 100.
4
3
6.2.4 Ordered Partitions
1. Ordered Partitions (distinguishable objects)
(a) The total number of ways of putting n distinct objects into r compartments which
n
are marked as 1, 2, ...... r is equal to r .
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 1 !n 2 ! ...... !
n
r
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