Show that among any 4 n−1 people, there are either some n of them who mutually know each other, or some n who mutually don’t know each other.

Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

A:

Q: Prove that Master Cheng has won precisely 21 times in a row during some consecutive days.

A: Please find the solution

Q: Suppose that in a class, there are n many boys and n many girls. Among these 2n many people, we will…

A: Let A be a set of n elements. To select r objects from a collection of n elements in the set A…

Q: Suppose that you know that a golfer plays the first hole of a golf course with an infinite number of…

A: To prove that the golfer plays every hole on the course when given that a golfer plays the first…

Q: Compute the sum of the first 60 positive integers that are exactly divisible by 4.

A: The numbers divisible by 4 is:

Q: How many five-digit numbers are there for which each digit is either equal to both adjacent ones, or…

A:

Q: Use the ertended pigeonhole principle to show that there is at least 15 ways to choose 4 integers…

A: Extended Pigeonhole Principle:      If there are n pigeons and m pigeonholes and n>m. Then, there…

Q: Suppose a set of 8 numbers are selected from the set: {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13;…

A: It is given that 8 numbers are selected from the set 1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: (a) Show that if seven integers are selected from the first 10 positive integers, there must be at…

A:

Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n>1. All prime divisors of n!+1 must be bigger than n. This is…

Q: Prove that for any odd positive integer, n, the sum of nconsecutive positive integers is divisible…

A: To prove for any odd positive integer, n, the sum of nconsecutive positive integers is divisible by…

Q: In how many distinct ways can the digits in the number 1,238,229,168 be arranged?

A: We have to find in how many ways the number  1,238,229,168 can be arranged.

Q: Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

A:

Q: Show that for all n:  ceil(n) = -floor(-n)

A: In the given question we have to prove that ceil(n)=-floor(-n) That is…

Q: Find the sum of the positive even integers up to and including 50.

A:

Q: Suppose that Hilbert’s Grand Hotel is fully occupied on the day the hotel expands to a second…

A: We know, Hilbert's Grand Hotel is a thought experiment which gives counterintuitive property of…

Q: How many ways are there to invite one of four different friends over for dinner on five successive…

A: Let x, y ,z, u be the friends and a, b, c, d denotes the number of times a friend is invited…

Q: Discrete

A:

Q: 4. Suppose we are in a parallel universe where 6 dollar coins and 11 dollar coins exist. Show that…

A:

Q: How many positive integers less than 10000 are there which contain at least one 2 or at least one 3…

A:

Q: How many different integers must be selected to guarantee that at least 5 of them would have the…

A: Dividing any integer by 3, can have remainder 0, 1 or 2. Therefore, possibilities for remainder are…

Q: Compute the sum of all integers between 40 and 100 that are exactly divisible by 3.

A: Compute the sum of all integers between 40 and 100 that are exactly divisible by 3.

Q: Show that if there are 8 people in a room, then at least two of them must have been born on the same…

A: There are 7 days in a week.  So we can have 7 people who can have different birthdays from Sunday to…

Q: Let A be the set of all positive integers whose last two digits form a number that is evenly…

A: We have the given condition as   Let us consider A be the set of all positive integers whose last…

Q: Prove that there is some positive integer n such that n, n + 1,n + 2,... , n + 200 are all…

A:

Q: prove that the one's digit of any integer,n, is the remainder when one divides n by 10.

A:

Q: If n > 1 is an integer not of the form 6k + 3, prove that n + 2" is composite. [Hint: Show that…

A: Congruence modulo relation on integers: Let a, b∈Z and let m be a fixed positive integer. Then we…

Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: Suppose there is a currency which only uses 3-cent coins and 7-cent coins. Prove that for all…

A:

Q: show that every integer greater than 11 is the sum of two composite integers

A: We need to prove that  every integer greater than 11 is the sum of two composite integers.

Q: Suppose we randomly select a subset of positive integers. How large must the subset be, if we want…

A: According to the given question, if any positive integer n belongs to the required subset then it…

Q: Prove that when you select n +1 numbers from {1,2, ... 2n} two of them must be consecutive (that is,…

A: Use inductive argument and Pigeon hole principles

Q: Show that if n is an odd integer then []

A:

Q: The sum of all positive integers less than 400 which are divisible by 5.

A: Positive integers less than 400 which are divisible by 5 are⇒n=79 5,10,15,20,...,395 The above…

Q: Use a direct proof to show that the sum of two odd integers(k and l) is even.

A: The number n is even if it is expressed as  n=2m , where m is an integer. The number n is odd if it…

Q: Use generating functions to find the number of ways tochoose a dozen bagels from three…

A: Consider the provided question, Each variety is present at least 2 times in the dozen bagels, the…

Q: Prove that out of 14 people either there are three, each pair of whom are acquainted, or there are…

A: Suppose there is a person A among the 14 people. Look at the following representation, where the…

Q: If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which…

A: The number of r combinations which can be selected from n numbers if repetitions allowed is…

Q: Show that in a group of people there must be two individuals that have the same number of friends…

A: We make a proof by contradiction and use the pigeonhole principle.

Q: 1. Show that if seven integers are selected from the first 10 positive integers, there must be at…

A: We can solve the following question by pigeonhole principle

Q: Prove that for any odd positive integer, n, the sum of n consecutive positive integers is divisible…

A:

Q: Show that in a group of five people (where any two people are either friends or enemies), there are…

A: Lets us take five hypothetical people, A  B  C  D  E Lets choose the first person A. Now in the left…

Q: Prove that among any group of people, there are always two people with the same number of friends…

A:

Q: How large a subset of the integers from 1 to 50 must be chosen to guarantee that one of the numbers…

A: We have to determine How large a subset of integers from 1 to 50 must be chosen to guarantee that…

Q: 2) How many elements are in the union of 5 sets if the sets contain 10,000 elements, each pair of…

A: Given that there are 5 sets and sets contains 10,000 elements, each pair of sets has 1000 elements,…

Q: 5!, called 5_______ , is the product of all positive integers from_________ down through________ .…

A: For any positive integer n, n! is the product of all the positive integers  from 1 to n. That is,…

Q: Suppose that 150 people celebrated their birthday in three months, where 65 people celebrated their…

A: Hello! As you have posted more than 3 sub parts, we are answering the first 3 sub-parts.  In case…

Q: Prove that the sum of two consecutive odd integers is divisible by 4.

A: Let the two consecutive integers be n and n+2 Then we need to prove that (n)+(n+2) is divisible by 4…