The document discusses combinations and restricted combinations. Some key points: - Combinations refer to the number of ways of selecting items without regard to order, as opposed to permutations which consider order. - A combination formula is given to calculate the number of combinations of n items taken r at a time. - Examples demonstrate calculating combinations in situations like selecting committee members or players for a team when certain items must or cannot be included. - The number of combinations when p particular items must be included is written as n-pCr-p, and when p items cannot be included is n-pCr.

1. Combinations

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

4. Combination Formula
!
!( )!
n
r
n
C
r n r


( )
n n
r n rC C 
0 1n n
nC C 
1
1
n n n
r r rC C C
 

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