Probability Explanation And Exercises Worksheet Page 16
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37picking up two pieces? Table 2 lists all the possibilities. The first choice can be any
of the four colors. For each of these 4 first choices there are 3 second choices.
Therefore there are 4 x 3 = 12 possibilities.
Table 2. Twelve Possible Orders.
Number
First
Second
1
red
yellow
2
red
green
3
red
brown
4
yellow
red
n
!
5
yellow
green
P
=
n
r
n
r
(
-
) !
6
yellow
brown
7
green
red
n!
8
green
yellow
P
=
n
9
green
brown
r
(n
r) !
-
10
brown
red
n
!
11
brown
yellow
P
=
!
4 x 3 x 2 x 1
n
4
r
(
n
-
r
) !
P
12
12
brown
green
=
=
=
4
2
(
) !
2 x 1
4
-
2
More formally, this question is asking for the number of permutations of four
n!
P
=
n
things taken two at a time. The general formula is:
r
(n
r) !
-
n!
P
=
n
r
(n
r) !
!
4 x 3 x 2 x 1
-
4
P
12
=
=
=
4
2
(
) !
2 x 1
4
-
2
where
P
is the number of permutations of n things taken r at a time. In other
n
r
!
4 x 3 x 2 x 1
4
n!
words, it is the number of ways r things can be selected from a group of n things.
P
12
=
=
=
P
=
n
r
4
2
(n
r) !
-
(
) !
2 x 1
4
-
2
In this case,
!
4 x 3 x 2 x 1
4
P
12
=
=
=
4
2
(
) !
2 x 1
4
-
2
n!
It is important to note that order counts in permutations. That is, choosing red and
C
=
n!
n
r
then yellow is counted separately from choosing yellow and then red. Therefore
(n
r) !
-
r
!
C
=
n
r
(n
r) !
permutations refer to the number of ways of choosing rather than the number of
-
r
!
200
!
4 x 3 x 2 x 1
4
C
=
=
=
6
4
2
4 x 3 x 2 x 1
(
) !
2 x 1
x
4!
4
-
2 2
!
(
)(
2 1
)
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