STAT 400
Examples for 09/01/2011
Fall 2011
Multiplication Principle (
Fundamental Rule of Counting
):
If there are
n
events and event
i
can occur in
N
i
possible ways, then
the number of ways in which the sequence of
n
events may occur is
N
1
⋅
N
2
⋅
...
⋅
N
n
1.
Manager of a radio station decided that every day the broadcast will start
with one of the 9 Beethoven Symphonies, followed by one of Mozart's
27 Piano Concertos, followed by one of
Schubert’s 15 String Quartets.
Approximately how many years can the station do that without repeating
the program?
9
•
27
•
15
=
3645
days
Beethoven
Symphony
Mozart's
Piano
Concerto
Schubert’s
String
Quartet
3645 days
≈
10
years.
1
½
.
The call letters of radio and television stations in the United States begin with either
K or W.
Those west of the Mississippi River start with K and those east of it with W.
a)
Some stations, such as KID in Idaho Falls, Idaho, and WOW in Omaha, Nebraska,
have 3call letters.
How many sets of call letters having 3 letters re possible?
2
×
26
×
26 =
1,352
.
b)
Most stations that were licensed after 1927 have 4 call letters, such as KUZZ in
Bakersfield, California, and WXYZ in Detroit, Michigan.
How many sets of call
letters having 4 letters are possible?
2
×
26
×
26
×
26 =
35,152
.
c)
How many sets of call letters having 4 letters are possible if we are not allowed to
repeat letters?
2
×
25
×
24
×
23 =
27,600
.
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In how many orders can the names of 5 candidates for the same office be
listed on a ballot?
5
•
4
•
3
•
2
•
1
=
120
.
1st
2nd
3rd
4th
5th
n
! = 1
⋅
2
⋅
...
⋅
(
n
– 1
)
⋅
n
n
! =
n
⋅
(
n
– 1
)
⋅
...
⋅
2
⋅
1
0
! = 1
n
! is the number of ways to rearrange (reorder)
n
distinct items.
For example, there are 7
! = 5,040 different ways to arrange 7 books on a bookshelf.
3.
How many ways are there of scrambling the letters of the word SCRAMBLE ?
There are 8 letters in the word SCRAMBLE, none of them repeating.
Therefore, there are 8
! =
40,320
different ways to rearrange the letters.
4.
Eight horses are entered in a race in which bets are placed on which horse
will win, place, and show (that is, finish first, second and third).
Suppose
that the race is run and there are no ties.
a)
In how many orders can all eight horses finish the race?
8
! =
40,320
different ways for eight horses to finish the race.
b)
In how many ways can the win, place, and show be taken?
8
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 Fall '08
 Kim
 Statistics, Counting, Probability, ice cream

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