2. Meaning of Combinations
We know that the total number of permutations of
two persons out of four persons A, B, C, D is 4P2 =
12. They are as follows:
AB AC AD
BA BC BD
CA CB CD
DA DB DC
Sometimes the persons are to be selected not
arranged. Two out of four persons A, B, C, D can be
selected in the following different ways:
AB, AC, AD, BC, BD, CD
3. Meaning of Combinations
Thus two persons out of four persons can be
selected in 6 different ways .
These different ways of selection are known as
Combinations.
AB and BA are two different permutations while in
combination the selection AB and BA is considered
only one.
Notation: 4C2 = 6
5. Ex.1 Find the values of the following:
1. 12C4
2. 25C23
3. 8C2
4. 11C7
6. Ex.2
In how many ways a committee of 4 professors can
be formed out of 11 professors?
7. Solution:
The total number of combinations of 4 professors out
of 11 professors
= 11C4
= 11! / ( 4! . 7!)
=_____
=330
8. Ex. 3
In how many ways 4 Gujaratis, 2 Punjabis and 1
Madrasi can be selected out of 8 Gujaratis, 4
Punjabis and 3 Madarasis?
9. Solution:
4 Gujaratis out of 8 can be selected in 8C4 ways.
2 Punjabis out of 4 can be selected in 4C2 ways
1 Madarasi out of 3 can be selected in 3C1 ways.
Therefore total number of combinations
= 8C4
x 4C2
x 3C1
=______
=1260
10. Restricted Combinations
The number of combinations of n things taken r at a
time in which p particular things always occur is n-pCr-
p.
p particular things never occur is n-pCr.
11. Ex. 4
In how many ways a legal committee of 6 members
can be formed out of 11 ministers of a cabinet so
that the chief minister and the minister of law and
order are included?
Moreover if two particular ministers are not to be
taken in the committee, in how many ways can it be
formed?
12. Solution:
Total 6 ministers are to selected out of 11.
If the chief minister and the minister of law and order
are to be included then remaining 4 ministers out of
9 can be selected in
1C1 x 1C1 x 9C4=____
=126
If two particular ministers are not to be taken in
committee, then 1 C.M, 1 minister of law and order
and 4 out of 7 (9-2=7) ministers can be selected in
1C1 x 1C1 x 7C4=_______
=35
13. Ex.5
A box contains 5 red, 3 green and 2 white balls. In
how many ways 3 balls can be drawn from it such
that
(i) one ball of each colour is included?
(ii) 2 balls of the same colour and 1 ball of different
colour is included?
(iii) 3 balls of the same colour are included?
14. Solution:
A box contains 5 red, 3 green and 2 white balls
(i) the number of ways of drawing one ball of each of
the three colours
= 5C1 x 3C1 x 2C1 = _______
=30
(ii) 2 balls of the same colour and 1 ball of different
colour can be draws in the following ways:
2 red balls and 1 from the remaining 5 balls OR
2 green balls and 1 from the remaining 7 balls OR
2 white balls and 1 from the remaining 8 balls
15. Therefore, the total number of combinations +
= 5C2 x 5C1 + 3C2 x 7C1 + 2C2 x 8C1
=________
=79
(iii) 3 balls of the same colour can be all the three
red balls or all the three green balls
Therefore the total number of ways
= 5C3 + 3C3
=___
=11
16. Ex.6
A cricket team consists of 15 players including 4
bowlers and 2 wicket keepers. In how many ways 11
players can be selected of them so that 3 bowlers
and 1 wicket-keeper are included in the team?
17. Solution:
Total 11 players are to be selected from 15.
3 bowlers out of 4, 1 wicket-keeper out of 2 and
other 7 players from the remaining 9 can be selected
in
= 4C3 x 2C1 x 9C7 = _______
=288