How many 98-element subsets does [100] have? (Here [100] is the set consisting of the first 100 positive integers). Hint: This should not be a lengthy computation.

Q: Let S be a set of non-negative integers with 80 elements. Show that we can find a subset of S such…

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Q: If positive integers are chosen at random, what is the minimum number you must have in order to…

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Q: What is the set Z+ of all positive integers?

A: Set of integers are represented as: ℤ={.....,-3,-2,-1,0,1,2,3.........} Set of natural numbers:…

Q: Suppose that the number of subsets with k elements, made out of a set of n elements is called by the…

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Q: Is the set of all of the integers between 12 and 99 finite or infinite?

A: Given : The set of all of the integers between 12 and 99

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: How many strings are there over the set {a, b, c} that have length 10 in which no two consecutive…

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Q: The set of numbers whose decimal representations are neither terminating nor repeating is called the…

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Q: Determine the number of permutations of the set {1 ,2 ,..., 14} in which exactly 7 integers are in…

A: Solution-:

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: Show that A5 has 24 elements of order 5, 20 elements of order 3, and15 elements of order 2.

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Q: Suppose that in a particular Finite course, 5/14 of students plan on majoring in math, 11/42 plan on…

A: Given information- We have given the proportion of particular Finite course. P (Major in math) =…

Q: Show that A5 has 24 elements of order 5, 20 elements of order 3, and 15 elements of order 2.

A: We know that A5 has only even order permutation. A5 has 5!2=60 elements. Now write the partition of…

Q: Determine the identity element of the set N={1,2,3,..} of positive integers with addition.

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Q: A(n) _____ is a collection of items called elements

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Q: How many elements of order 5 are in S7?

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Q: The number of substrings (of all lengths inclusive) that can be formed from a character string of…

A: A string is a set of characters. A substring is a set of characters that can be formed by taking 1…

Q: Discrete

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Q: How many pairs of integers (a,b) exists such that the product of a,b and HCF of (a,b) is 1080??

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Q: How many elements are there in the greatest conjugacy class of 5 ?S

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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: Find the order of [5] in Z23

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Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

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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: Refer to a set of five distinct history books, three distinct biology books, and two distinct…

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Q: Let X = {x;} be a set of n positive integers. Show that the sum of the integers of a subset of X is…

A: It is required to show that sum of elements of a subset of described set, X, is divisible by integer…

Q: Consider lists made from L, a, T, e, X, with repetitions allowed. How many 4-lists are there in…

A: Please refer to the image below.

Q: Show that given any set of 5 natural numbers, there are 3 numbers whose sum is divisible by 3.

A: Let q∈N. Note that, for every q there exists some x∈N and some v∈0,1,2 such that q=3x+v. Now, let…

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: how many subsetsof one or more elemets can be formed from a set containing 12 elements?

A: Use concept of power set

Q: Find how many positive integers with exactly four decimal digits, that is, positive integers between…

A: Note : According to our guidelines we can solve only first three subparts and rest can be…

Q: Show that there are infinitely many integers n such that 4(n^2)+1 is simultaneously divisible by 13…

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Q: 20 The set [20] contains = = 15504 five-element subsets. How many of them contain more odd elements…

A: There are 15504 five element subsets. The set of 20 numbers contains 10 odd and 10 even numbers. If…

Q: Let S be the set of all (nonnegative) 4-digit integers. How many integers does S contain?

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Q: [Q1] From the letters {A, B, C, D, E, F, G}, how many: (a) 3-distinct-letters co can be generated?…

A: Given: There are 7 letters A, B, C, D, E, F, G

Q: Let A denote the set of all 4 digit numbers that are divisible by 3, and B denote the set of all 4…

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Q: If n is an integer greater than 2, then for all integers m, n does not divide m or n + m # nm.

A: c Given n∈Zand n>2. We have to prove that for all m∈Z, n/m or n+m/nm. Suppose n/m and…

Q: Let S be the set of all (nonnegative) 4-digit integers. How many integers does S contain? Enter your…

A: Disclaimer: Since you have asked multiple questions we will solve the first question for you. If you…

Q: How many substrings can be formed from a character string of length n?

A: Introduction :

Q: How many integers between 100 and 999 have distinct digits, and how many of those are even numbers?

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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 the set of odd numbers is countable.

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Q: that the set of negative integers divisible by 3 is countable

A: Set of negetivee integers divisible by 3 isS=-3n, n∈ℕf: ℕ→S   defined asf(n) = -3n,To check…

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: Show that in any collection of 7 or more natural numbers there are ALWAYS TWO numbers whose sum or…

A: In the given problem we have to show that in any collection of 7 or more natural numbers there are…

Q: In N, where N is a set of natural numbers, multiplication by '.' Identify the reversible elements.

A: Query-  In N, where N is a set of natural numbers, multiplication by '.' To Identify- The reversible…

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…

Q: Consider strings of lowercase letters. How many strings are there of lowercase letters of length…

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