Chapter 10 Study
If only the digits 0, 3, 4, 5, and 9 may be used, find the number of possible three-digit numbers that can be formed.
100
What is the least four-digit number in the first 15 rows of Pascal's triangle?
1001
In 5-card poker, a straight is five cards of consecutive denominations (not all the same suit). How many ways are there to get a straight?
10200
If you toss 10 fair coins, in how many ways can you obtain at least one head and one tail?
1022
In how many ways could members of the following club line up all 12 members for a photograph? N={Carl, Lisa, Jen, Adam, Jenny, Bill, Tami, Kate, Joe, Alan, Ted, Carlos}
12! = 479001600
f you toss 7 fair coins, in how many ways can you obtain at least two heads? ("At least two" is the complement of "zero or one.")
120
Determine the total number of proper subsets of the set of letters from the English alphabet {a, b, c, ... , g}.
127
How many proper subsets are there of the set {K, L, M, N, O, P, Q}.
127
A customer ordered fourteen zingers. Zingers are placed in packages of four, three, or one. In how many different ways can this order be filled?
13
Phil has 6 books to read, but cannot take them all on vacation. Use Pascal's triangle to find the number of ways he can choose 2 books
15
Suppose 6 fair coins are tossed . Use Pascal's triangle to find the number of ways of obtaining exactly 2 heads.
15
Determine the number of permutations (arrangements) of the following. 10 objects taken 6 at a time
151200
A panel containing four on-off switches in a row is to be set. Assuming no restrictions on individual switches, use the fundamental counting principle to find the total number of possible panel settings.
16
The table to the right categorizes 20 senators as to political party and gender. One member is chosen at random. In how many ways can the chosen person be a woman or a Republican man?
18
Verify that 17C7 = 17C10
19448 for both Yes
The table to the right categorizes 30 senators as to political party and gender. One member is chosen at random. In how many ways can the chosen person be a woman or Democrat?
21 senators that are a woman or a democrat
How many ways can a president, vice-president, and secretary be chosen from a committee of 7 people?
210
How many ways can a male and a female be selected to decorate for a party from a club consisting of ten members where four are men and six are women?
24
A committee of eight Congressmen will be selected from a group of seven Democrats and nine Republicans. Find the number of ways of obtaining exactly one Democrat.
252
For a set of three objects, find the number of different subsets of size 1. (Use row 3 of Pascal's triangle to find the answer.)
3
How many different ways could four distinct days of the week be chosen so that at least one of them begins with the letter S?
30
A certain restaurant offers 5 choices in the soup and salad category (3 soups and 2 salads), 2 choices in the bread category, and 6 choices in the entree category. Find the number of dinners available if one soup, one salad, and one entree are to be included.
36 dinner combinations including one soup, one salad, and one entree?
In 5-card poker, a royal flush is an ace, king, queen, jack, and 10, all of the same suit. How many ways are there to get a royal flush?
4
Four friends board an airliner just before departure time. There are only 10 seats left, 3 of which are aisle seats. How many ways can the 4 people arrange themselves in available seats so that at least one of them sits on the aisle?
4200
An electronics store receives a shipment of 20 graphing calculators, including 8 that are defective. Four of the calculators are selected to be sent to a local high school. How many of these selections will contain no defective calculators?
495
For a set of five elements find the number of different subsets of 4 elements. Use row 5 of Pascal's triangle to find the answer.
5
In 5-card poker, three of a kind is 3 cards of one denomination, plus 2 cards of two additional denominations. How many ways are there to get three of a kind?
54912
A doughnut shop has a special on its Mix-n-Match selection, which allows customers to select three doughnuts from the following varieties: Boston creme, frosted, apple cider, maple, chocolate, and glazed. How many different Mix-n-Match selections are possible? [Hint: Consider when all three doughnuts are the same, two are the same, and none are the same.]
56
Find the number of combinations (subsets) of 8 things taken 3 at a time.
56
An unusual die has the numbers 1, 1, 2, 2, 6, and 6 on its six faces. Two of these dice are rolled, and the two numbers on the top faces are added. How many different sums are possible?
6
A committee of eight Congressmen will be selected from a group of thirteen Republicans and seven Democrats. Find the number of ways of obtaining a committee with exactly three Republicans.
6006
Determine the number of permutations (arrangements) possible of 10 things taken 2 at a time.
90
In how many ways could a club select two members, one to open their next meeting and one to close it, given that Alan will not be present? N={James, Sandy, Jane, Jenny, Bill, Tami, Kate, Joe, Alan, Ted, Carlos}
90 ways
A system of identifying call letters uses call letters that begin with either R or I. Some have a total of four letters, and others have five letters. How many different call letter combinations are possible? Count all possibilities even though, practically, some may be inappropriate. (Hint: Do not apply combinations.)
949104
How many subsets (of any size) are there for a set of zero elements?
A set of zero elements has 1 subset
A club N with four members is shown below. N={Adam, Blake, Cassie, Daniel}, abbreviated as N={A, B, C, D} Assuming all members of the club are eligible, but that no one can hold more than one office, list and count the different ways the club could elect both a president and a treasurer.
AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC 12 Ways
Assuming all members of the club are eligible, but that no one can hold more than one office, list and count the different ways the club could elect a president and a treasurer if the two officers must not be the same gender. N={Aaron, Ben, Cassie, Dennis, Emma} or, in abbreviated form, N={A, B, C, D, E}.
AC, AE, BC, BE, CA, CB, CD, DC, DE, EA, EB, ED 12 Ways
A club is made up of the members listed below. Assuming that all members are eligible, but no one can hold more than one office, list and count the different ways the club could elect a president and a treasurer if the president must be a woman. (Cameron and Eileen are women, and the others are men.) N={Aaron, Ben, Cameron, Derek, Eileen} or, in abbreviated form N={A, B, C, D, E}. List the different ways the club could elect each group of officers, where the first letter represents the president's name and the second letter represents the treasurer's name. Choose the correct answer below.
CA, CB, CD, CE, EA, EB, EC, ED 8 Ways
A contractor builds homes of 7 different models and presently has 3 lots to build on. In how many different ways can he arrange homes on these lots? Assume 3 different models will be built.
Different Arrangements = Permutations 210 arrangements
Refer to the table below. Of the 36 possible outcomes, determine the number for which the sum (for both dice) is 2
One can roll a sum of 2 in 1 way(s)
Counting numbers are to be formed using only the digits 4, 8, and 9. Determine the number of different possibilities for the type of number described below. Four-digit numbers with one pair of adjacent 8s and no other repeated digits (Hint: You may want to split the task of designing such a number into three parts, such as (1) position the pair of 8s, (2) position the 4, and (3) position the 9.)
The number of different possibilities for this type of number is 6
The figure to the right shows a portion of Pascal's triangle with several inverted triangular regions outlined. For any one of these regions, what can be said of the sum of the squares of the entries across its top row?
The sum of the squares of the entries across the top row equals the entry at the bottom vertex.
Refer to the table below. Of the 36 possible outcomes, determine the number for which the sum (for both dice) is odd
There are 18 outcomes where the sum is odd
Refer to the table below. Of the 36 possible outcomes, determine the number for which the sum (for both dice) is greater than 10
There are 3 ways that the sum can be greater than 10
Determine whether the object is a permutation or a combination.
This is a combination because repetition is not allowed and the order of the items does not matter.
Determine whether the object is a permutation or a combination. a 10-digit telephone number (including area code)
This is neither a permutation nor a combination because repetition is allowed.
Ann's collection of twelve albums includes one jazz album. Ann will choose three of her albums to play on a road trip. (Assume order is not important.) a) How many different sets of three albums could she choose? b) How many of these sets would not include the jazz album? c) How many of them would include the jazz album?
a. 220 b. 165 c. 55
Jason wants to dine at four different restaurants during a summer getaway. If four of nine available restaurants serve seafood, find the number of ways that at least one of the selected restaurants will serve seafood given the following conditions. (a) The order of selection is important. (b) The order of selection is not important.
a. 2904 b. 121
(the contractor) is to build seven homes on a block in a new subdivision, using two different models: standard and deluxe. (All standard homes are the same, and all deluxe models are the same.) (a) How many different choices does Leo have in positioning the seven houses if he decides to build five standard and two deluxe models? (b) If Leo builds four deluxes and three standards, how many different positionings can he use?
a. Leo has 21 choices in positioning the seven houses b. If Leo builds four deluxes and three standards, he can use 35 different positionings
