Chapter 10
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1
Counting Techniques
2
4
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Combinations
Section 10.3
1
2
4
Combinations
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 A selection of distinct objects
1
without regard to order is a
combination.
2
4
Combination Formula
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 The number of combinations of n
objects, taken r at a time(order is
not important and n  r).
1
C

n
r
n!
( n  r )! r !
2
4
Combination Formula
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 The number of combinations of n
objects, taken r at a time(order is
not important n  r).
1
C

n
r
P
n r
r!
2
4
Combination Rule
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 How many ways can 3 cards be chosen
from a standard deck of 52 cards,
disregarding the order of the selection?
1
2
4
52 nCr 3 = 52 x 51 x 50 = 22,100
3x2x1
Combination Rule
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If 20 people all shake hands with each other, how
many handshakes are there?
20 nCr 2 = 20 x 19 = 190
2
1
2
The Greek alphabet has 24 letters. In how many ways can 3
different Greek letters be selected if the order does not matter?
24 nCr 3 = 24 x 23 x 22 = 2024
3x2x1
4
Combination Rule
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A committee is to consist of 3 members. If there
are 4 men and 6 women available to serve on
this committee, find the following:
1
2
a. How many different committees can be formed?
10 x 9 x 8 = 120
10 nCr 3 =
3x2x1
4
b. How many committees can be formed if each
committee must consist of 2 men and 1 woman?
4 nCr 2 x 6 nCr 1 = 6 x 6 = 36
Combination Rule
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How many different committees can be
formed from 8 people if each committee
must consist of at least 3 people?
1
2
8 nCr 3 + 8 nCr 4 + 8 nCr 5 + 8 nCr 6 + 8 nCr 7 + 8 nCr 8 =
4
56 + 70 + 56 + 28 + 8 + 1 = 219
Combination Rule
How many committees of 5 people can be
formed from 9 men and 7 women if the
committee must consist of less than 3 men?
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Determine what is acceptable for each
gender in order to have a committee of five.
Acceptable
Men
Women
0
5
1
4
2
3
1
Solution:
2
9 nCr 0  7 nCr 5 + 9 nCr 1  7 nCr 4 +9 nCr 2  7 nCr 3
4
121 + 935 + 3635
21 + 315 + 1260
1596
Combination Rule
How many committees of 6 people can be
formed from 9 men and 7 women if the
committee must consist of more than 4
women?
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Determine what is acceptable for each
gender in order to have a committee of six.
Acceptable
Men
Women
1
5
0
6
Notice 7 is not acceptable for the women.
Solution:
1
2
4
9 nCr 1  7 nCr 5 + 9 nCr 0  7 nCr 6
921 + 17
189 + 7
196
END