1.)Eight horses are entered in a race.
(a) How many different orders are possible for completing
the race?
(b) In how many different ways can first, second, and third
places be decided? (Assume there is no tie.)
2.)Telephone numbers consist of seven digits; the first digit
cannot be 0 or 1. How many telephone numbers are
possible?
3.)In how many ways can five people be seated in a row of
five seats?
4.)In how many ways can five different mathematics books be
placed next to each other on a shelf?
5.)In a family of four children, how many different boy-girl
birth-order combinations are possible? (The birth orders
BBBG and BBGB are different.)
6.)Two cards are chosen in order from a deck. In how many
ways can this be done if
(a) the first card must be a spade and the second must be a
heart?
(b) both cards must be spades?
7.)A company’s employee ID number system consists of one
letter followed by three digits. How many different ID
numbers are possible with this system?
8.)An all-star baseball team has a roster of seven pitchers and
three catchers. How many pitcher-catcher pairs can the
manager select from this roster?
9.)An automobile dealer offers five models. Each model
comes in a choice of four colors, three types of stereo
equipment, with or without air conditioning, and with or
without a sunroof. In how many different ways can a
customer order an auto from this dealer?
10.)How many monograms consisting of three initials are
possible?
11.)A state license plate design has six places. Each plate
begins with a fixed number of letters, and the remaining
places are filled with digits. (For example, one letter
followed by five digits, two letters followed by four
digits, and so on.) The state has 17 million registered
vehicles.
(a) The state decides to change to a system consisting of
one letter followed by five digits. Will this design
allow for enough different plates to accommodate all
the vehicles registered?
(b) Find a system that will be sufficient if the
smallest possible number of letters is to
be used.
12.)In how many ways can a president, vice president, and secretary
be chosen from a class of 20 females and 30 males if
the president must be a female and the vice president a
male?
13.)Social Security numbers consist of nine digits, with the first
digit between 0 and 6, inclusive. How many Social Security
numbers are possible?
14.)How many five-letter palindromes are possible? (A palindrome
is a string of letters that reads the same backward
and forward, such as the string XCZCX.)
15.)How many different three-character code words consisting
of letters or digits are possible for the following code
designs?
(a) The first entry must be a letter.
(b) The first entry cannot be zero.
16.)Three-digit numbers are formed using the digits 2, 4, 5, and
7, with repetition of digits allowed. How many such numbers
can be formed if
(a) the numbers are less than 700?
(b) the numbers are even?
(c) the numbers are divisible by 5?
17.)Until recently, telephone area codes in the
United States, Canada, and the Caribbean islands were
chosen according to the following rules: (i) The first digit
cannot be 0 or a 1, and (ii) the second digit must be a 0 or a
1. But in 1995, the second rule was abandoned when the
area code 360 was introduced in parts of western Washington
State. Since then, many other new area codes that violate
Rule (ii) have come into use, although Rule (i) still
remains in effect.
(a) How many area code telephone number combinations
were possible under the old rules? (See Exercise 6
for a description of local telephone numbers.)
(b) How many area code telephone number combinations
are now possible under the new rules?
(c) Why do you think it was necessary to make this
change?
(d) How many area codes that violate Rule (ii) are you personally
familiar with?