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statistics

Questions and Answers of Statistics

How many bit strings with length not exceeding n, where n is a positive integer, consist entirely of 1s, not counting the empty string?
How many strings are there of lowercase letters of length four or less, not counting the empty string?
How many strings of five ASCII characters contain the character @ ("at" sign) at least once?
How many 6-element RNA sequences a) Do not contain U? b) End with GU? c) Start with C? d) Contain only A or U?
How many positive integers between 50 and 100 a) Are divisible by 7? Which integers are these? b) Are divisible by 11? Which integers are these? c) Are divisible by both 7 and 11? Which integers are
How many strings of three decimal digits a) Do not contain the same digit three times? b) Begin with an odd digit? c) Have exactly two digits that are 4s?
A committee is formed consisting of one representative from each of the 50 states in the United States, where the representative from a state is either the governor or one of the two senators from
How many license plates can be made using either two uppercase English letters followed by four digits or two digits followed by four uppercase English letters?
How many license plates can be made using either two or three uppercase English letters followed by either two or three digits?
How many strings of eight English letters are there a) That contain no vowels, if letters can be repeated? b) That contain no vowels, if letters cannot be repeated? c) That start with a vowel, if
How many one-to-one functions are there from a set with five elements to sets with the following number of elements? a) 4 b) 5 c) 6 d) 7
How many functions are there from the set {1, 2, . . . , n}, where n is a positive integer, to the set {0, 1} a) That are one-to-one? b) That assign 0 to both 1 and n? c) That assign 1 to exactly one
How many partial functions (see Definition 13 of Section 2.3) are there from a set with m elements to a set with n elements, where m and n are positive integers?
Apalindrome is a string whose reversal is identical to the string. How many bit strings of length n are palindromes?
How many 4-element RNA sequences a) Contain the base U? b) Do not contain the sequence CUG? c) Do not contain all four bases A, U, C, and G? d) Contain exactly two of the four bases A, U, C, and G?
How many ways are there to seat six people around a circular table where two seatings are considered the same when everyone has the same two neighbors without regard to whether they are right or left
In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if a) The bride must be next to the groom? b) The bride is not next to the groom? c) The
How many bit strings of length 10 either begin with three 0s or end with two 0s?
Six different airlines fly from New York to Denver and seven fly from Denver to San Francisco. How many different pairs of airlines can you choose on which to book a trip from NewYork to San
How many positive integers not exceeding 100 are divisible either by 4 or by 6?
Suppose that a password for a computer system must have at least 8, but no more than 12, characters, where each character in the password is a lowercase English letter, an uppercase English letter, a
The name of a variable in the JAVA programming language is a string of between 1 and 65,535 characters, inclusive, where each character can be an uppercase or a lowercase letter, a dollar sign, an
Suppose that at some future time every telephone in the world is assigned a number that contains a country code 1 to 3 digits long, that is, of the form X, XX, or XXX, followed by a 10-digit
A wired equivalent privacy (WEP) key for a wireless fidelity (WiFi) network is a string of either 10, 26, or 58 hexadecimal digits. How many different WEP keys are there?
Use the principle of inclusion-exclusion to find the number of positive integers less than 1,000,000 that are not divisible by either 4 or by 6.
How many ways are there to arrange the letters a, b, c, and d such that a is not followed immediately by b?
Use a tree diagram to determine the number of subsets of {3, 7, 9, 11, 24} with the property that the sum of the elements in the subset is less than 28.
a) Suppose that a popular style of running shoe is available for both men and women. The woman's shoe comes in sizes 6, 7, 8, and 9, and the man's shoe comes in sizes 8, 9, 10, 11, and 12. The man's
How many different three-letter initials can people have?
Use mathematical induction to prove the sum rule for m tasks from the sum rule for two tasks.
How many diagonals does a convex polygon with n sides have? (Recall that a polygon is convex if every line segment connecting two points in the interior or boundary of the polygon lies entirely
How many different three-letter initials are there that begin with an A?
Show that in any set of six classes, each meeting regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no classes are held on weekends.
Let (xi, yi, zi), i = 1, 2, 3, 4, 5, 6, 7, 8, 9, be a set of nine distinct points with integer coordinates in xyz space. Show that the midpoint of at least one pair of these points has integer
a) Show that if five integers are selected from the first eight positive integers, there must be a pair of these integers with a sum equal to 9. b) Is the conclusion in part (a) true if four integers
How many numbers must be selected from the set {1, 2, 3, 4, 5, 6} to guarantee that at least one pair of these numbers add up to 7?
A company stores products in a warehouse. Storage bins in this warehouse are specified by their aisle, location in the aisle, and shelf. There are 50 aisles, 85 horizontal locations in each aisle,
Suppose that every student in a discrete mathematics class of 25 students is a freshman, a sophomore, or a junior. a) Show that there are at least nine freshmen, at least nine sophomores, or at least
Construct a sequence of 16 positive integers that has no increasing or decreasing subsequence of five terms.
Show that whenever 25 girls and 25 boys are seated around a circular table there is always a person both of whose neighbors are boys.
Show that in a group of 10 people (where any two people are either friends or enemies), there are either three mutual friends or four mutual enemies, and there are either three mutual enemies or four
Show that if n is an integer with n ≥ 2, then the Ramsey number R(2, n) equals n. (Recall that Ramsey numbers were discussed after Example 13 in Section 6.2.)
Adrawer contains a dozen brown socks and a dozen black socks, all unmatched. A man takes socks out at random in the dark. a) How many socks must he take out to be sure that he has at least two socks
Show that there are at least six people in California (population: 37 million) with the same three initials who were born on the same day of the year (but not necessarily in the same year). Assume
In the 17th century, there were more than 800,000 inhabitants of Paris. At the time, it was believed that no one had more than 200,000 hairs on their head. Assuming these numbers are correct and that
There are 38 different time periods during which classes at a university can be scheduled. If there are 677 different classes, how many different rooms will be needed?
Acomputer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in the network that are directly
Find the least number of cables required to connect 100 computers to 20 printers to guarantee that 2every subset of 20 computers can directly access 20 different printers. (Here, the assumptions
An arm wrestler is the champion for a period of 75 hours. (Here, by an hour, we mean a period starting from an exact hour, such as 1 p.m., until the next hour.) The arm wrestler had at least one
Show that if f is a function from S to T, where S and T are nonempty finite sets and m = [|S| / |T|], then there are at least m elements of S mapped to the same value of T. That is, show that there
An alternative proof of Theorem 3 based on the generalized pigeonhole principle is outlined in this exercise. The notation used is the same as that used in the proof in the text. a) Assume that ik
Show that among any group of five (not necessarily consecutive) integers, there are two with the same remainder when divided by 4.
Let n be a positive integer. Show that in any set of n consecutive integers there is exactly one divisible by n.
What is the minimum number of students, each of whom comes from one of the 50 states, who must be enrolled in a university to guarantee that there are at least 100 who come from the same state?
List all the permutations of {a, b, c}.
How many bit strings of length 10 contain a) Exactly four 1s? b) At most four 1s? c) At least four 1s? d) An equal number of 0s and 1s?
A group contains n men and n women. How many ways are there to arrange these people in a row if the men and women alternate?
In how many ways can a set of five letters be selected from the English alphabet?
How many subsets with more than two elements does a set with 100 elements have?
A coin is flipped 10 times where each flip comes up either heads or tails. How many possible outcomes a) Are there in total? b) Contain exactly two heads? c) Contain at most three tails? d) Contain
How many permutations of the letters ABCDEFG contain a) The string BCD? b) The string CFGA? c) The strings BA and GF? d) The strings ABC and DE? e) The strings ABC and CDE? f) The strings CBA and BED?
How many ways are there for eight men and five women to stand in a line so that no two women stand next to each other?
A club has 25 members. a) How many ways are there to choose four members of the club to serve on an executive committee? b) How many ways are there to choose a president, vice president, secretary,
How many 4-permutations of the positive integers not exceeding 100 contain three consecutive integers k, k + 1, k + 2, in the correct order a) Where these consecutive integers can perhaps be
How many permutations of {a, b, c, d, e, f, g} end with a?
The English alphabet contains 21 consonants and five vowels. How many strings of six lowercase letters of the English alphabet contain a) Exactly one vowel? b) Exactly two vowels? c) At least one
Suppose that a department contains 10 men and 15 women. How many ways are there to form a committee with six members if it must have the same number of men and women?
How many bit strings contain exactly eight 0s and 10 1s if every 0 must be immediately followed by a 1?
How many bit strings of length 10 contain at least three 1s and at least three 0s?
How many license plates consisting of three letters followed by three digits contain no letter or digit twice?
Find a formula for the number of circular r-permutations of n people.
How many ways are there for a horse race with three horses to finish if ties are possible?
There are six runners in the 100-yard dash. How many ways are there for three medals to be awarded if ties are possible? (The runner or runners who finish with the fastest time receive gold medals,
Find the value of each of these quantities. a) P(6, 3) b) P(6, 5) c) P(8, 1) d) P(8, 5) e) P(8, 8) f) P(10, 9)
How many possibilities are there for the win, place, and show (first, second, and third) positions in a horse race with 12 horses if all orders of finish are possible?
Give a formula for the coefficient of xk in the expansion of (x2 − 1/x)100, where k is an integer.
What is the row of Pascal's triangle containing the binomial coefficients
Show that if n and k are integers with 1 ‰¤ k ‰¤ n, then
Prove Pascal's identity, using the formula for
Prove that if n and k are integers with 1 ‰¤ k ‰¤ n, thena) Using a combinatorial proof. b) Using an algebraic proof based on the formula for (nr) given in Theorem 2 in
Prove the identity (nr)(rk) = (nk) (n-kr -k), whenever n, r, and k are nonnegative integers with r ≤ n and k ≤ r, a) Using a combinatorial argument. b) Using an argument based on the formula for
Let n be a positive integer. Show that
Find the expansion of (x + y)6.
In how many different ways can five elements be selected in order from a set with three elements when repetition is allowed?
How many ways are there to choose eight coins from a piggy bank containing 100 identical pennies and 80 identical nickels?
A book publisher has 3000 copies of a discrete mathematics book. How many ways are there to store these books in their three warehouses if the copies of the book are indistinguishable?
Suppose that a large family has 14 children, including two sets of identical triplets, three sets of identical twins, and two individual children. How many ways are there to seat these children in a
How many ways are there to distribute six indistinguishable balls into nine distinguishable bins?
How many ways are there to distribute 12 distinguishable objects into six distinguishable boxes so that two objects are placed in each box?
How many positive integers less than 1,000,000 have the sum of their digits equal to 19?
There are 10 questions on a discrete mathematics final exam. How many ways are there to assign scores to the problems if the sum of the scores is 100 and each question is worth at least 5 points?
How many different bit strings can be transmitted if the string must begin with a 1 bit, must include three additional 1 bits (so that a total of four 1 bits is sent), must include a total of 12 0
How many strings of six letters are there?
How many different strings can be made from the letters in ABRACADABRA, using all the letters?
How many different strings can be made from the letters in ORONO, using some or all of the letters?
How many strings with seven or more characters can be formed from the letters in EVERGREEN?
A student has three mangos, two papayas, and two kiwi fruits. If the student eats one piece of fruit each day, and only the type of fruit matters, in how many different ways can these fruits be
How many ways are there to travel in xyz space from the origin (0, 0, 0) to the point (4, 3, 5) by taking steps one unit in the positive x direction, one unit in the positive y direction, or one unit
How many ways are there to deal hands of seven cards to each of five players from a standard deck of 52 cards?
How many ways are there to deal hands of five cards to each of six players from a deck containing 48 different cards?

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