How do I prove that 1 is the smallest natural number by using Axiom of Induction & the well ordering principle

Q: One thousand four hundred and thirty-two is the 6th  permutation of the lexicographic permutations…

A: Fix the first element, {1}: (Similarly fix the next element and obtain the two permutations for the…

Q: What is the critical difference between proof by weak induction and proof by strong induction?

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Q: Eight people attend a meeting together and sit in a row with exactly eight seats that have passage…

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Q: Pigeonhole Principle Prove that among any 100 integers there always two whose difference is…

A: The given problem is to prove the following statement which follows the pigeonhole principle, we…

Q: How many 3-digit number greater than 200 can be formed (a) repetition not allowed (b) repetition…

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Q: Use the definition of an r-permutation, or the product rule to de number of trifectas possible in a…

A: Given: To explain the given statement as follows:

Q: Determine the number of permutations (arrangements) possible of 9 things taken 3 at a time

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Q: ive distinct integers are to be chosen from the set {1,2,3,4,5,6,7,8} and placed in some order in…

A: Given: To find the number of possible ways in which integers can be chosen,

Q: How many strings of Length tree can be formed by using the letter A,B,C,D and E without repetitions

A: Detailed explanation mentioned below

Q: 4. One way to add up n natural numbers can be described as: ("+1) which is in 'combinations' form.…

A: Generally sum of n natural numbers by using combinations we can find them using n+1c2. By using this…

Q: Prove by induction that it is possible to pay, without requiring change, any whole number of…

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Q: induction that

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Q: 5. Bayes' theorem provides a way to

A: #Bayes' theorem used in case of conditional probability i.e when prior probability of events are…

Q: A finite sequence of numbers is called AWESOME if each subsequent member of the sequence, except the…

A: Find the product of already given natural numbers. To make the product of all four natural numbers a…

Q: Let n and k be positive integers with 2k + 1 ≤ n. Find the number of ways to choose k children from…

A: Consider the given: Let n and k be positive integers with . Choose the number of ways in which k…

Q: (a) that the gap between the largest integer and the second largest one is at least 3. Show all…

A: Solution: Given The number of ways to select six integers from 1,2,..., n such that the gap between…

Q: Assume that live men and five women are to sit in a row of ten chairs. In how many ways can these…

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Q: Use permutations, combinations, the fundamental counting principle, or other counting methods as…

A: Here In the given there should be 20 students not 12 students since 3+6+11= 20

Q: 2. even numbers are in their natural positions. Determine the number of permutations of {1, 2, 3, 4,…

A: Here we use combination rules. And set theoretic rules.

Q: 89. Selecting Prizewinners From a group of 30 contestants, six are to be chosen as semifinalists,…

A: To find the number of ways we use combination  There are total of 30 contestants 6 are chosen as…

Q: How many positive integers less than 1000 are divisible by 5? How many positive integers less than…

A: We will answer the first three questions as per our guideline. Please resubmit the other question…

Q: The password consists of alphanumeric characters (i.e. letters or numbers) that distinguish between…

A: Introduction: According to the combination rule, if r distinct items are to be chosen from n items…

Q: How does Principle of Mathematical Induction works?

A: Principle of Mathematical induction is a technique which is used to obtain a mathematical proof of a…

Q: In a group of 23 people each person chooses a number between 1 and 20. Use the pigeon hole principle…

A: See the attachment

Q: A password a string that consists of 4 digits from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, followed…

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Q: Determine the probability that the randomly selected number from the interval [0,1] is a rational…

A: To Determine: The probability that the randomly selected number from the interval [0,1] is a…

Q: 4. Find the largest number of points which a foot ball team cannot get exactly using just 3-point…

A: For our company rules I solve only first question. Please repost other parts.

Q: Use mathematical induction to prove: For each natural number n, 6 divides (n3-n). I need to write a…

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Q: A person ordering a certain model of car can choose any of 9 colors, either manual or automatic…

A: A person is ordering a car, such that  there are 9 possible ways to choose colors, two ways to…

Q: A PIN-number for a credit card consists of a sequence of 4 digits chosen from the set…

A: Given the set {0,1,2,3,4,5,6,7,8,9} We find the 4 digits different PIN number

Q: 3. Explain the relationship among the three concepts: well ordered principle, induction, strong…

A: We know. Well ordered principle:  A set of numbers is known as well ordered when each of its…

Q: what the mean of superposition principle in math

A: According to the given information it is required to explain the superposition principle in maths.…

Q: Five distinct integers are to be chosen from the set {1, 2, 3, 4, 5, 6, 7, 8} and placed in some…

A: Given Information -  Five distinct integers are to be chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}…

Q: Using the pigeonhole principle How many numbers must be selected from the set {1, 3, 5, 7, 9, 11,…

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Q: A student in COBU260 course scores 110 points out of a possible 110 in the first half of the…

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Q: ) How many 4 digit numbers that are divisible by 10 can be formed from the numbers 3, 5, 7, 8, 9, 0…

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Q: students be invited, then compute the number of ways undergraduate students are invited.

A: Here use combination

Q: set of two - digit numbers obtained from the digits 1

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Q: 5 numbers are to be chosen from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} How many ways can 5 (not necesarilly…

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Q: A lottery game requires that you pick 7 different numbers from 1 to 59. If you pick all 7 winning…

A: According to the combination rule, if r distinct items are to be chosen from n items without…

Q: Let n be a positive integer. Give a recursive algorithm for computing the sum of the first n…

A: To find - Let n be a positive integer. Give a recursive algorithm for computing the sum of the first…

Q: Prove that the First Principle of Mathematical Induction is a consequenceof the Well Ordering…

A: Well ordering principle: Every non-empty set of positive integers contains a least element. In other…

Q: How many words can be formed by permutation of the letters of the word DEPRESSION such that each…

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Q: How many positive integers less than 2ˆ{20} are not divisible by 7, 11 or 13. Use…

A: The solution is given below

Q: Define Permutation and Combination in your own wo

A: Permutation is a way to arrange the given objects or numbers in order .

Q: ________________ draws from the box, there is more than 50% chance for a red marble to appear at…

A: here given box contains 2 red marbles and 98 blue ones so total 100 marbles  Here Selection is…

Q: How many twin primes are there where both primes are less than or equal to 31? Number of twin…

A: Twin prime: If there is only one composite number between two prime numbers, then that two numbers…

Q: With a short time remaining in the​ day, a delivery driver has time to make deliveries at 5…

A: Given information- We have given that a delivery driver has time to make deliveries at 5 locations…

Q: How many natural number that are less or equal to 1000 and that are not multiplea of 3,5or7 exist

A: To find: Number of natural numbers ≤1000 that are not multiple of 3 or 5 or 7.