(a) that the gap between the largest integer and the second largest one is at least 3. Show all Determine the number of ways to select six integers from 1,2,...,n such your work.

Q: How many compositions of n are there into k parts of size at least 3?

A: How many compositions of n are there into k parts of size at least 3?

Q: Prove that the hundreds digit of n20 is even.

A: To Prove that the hundreds digit of n20 is even.

Q: Show that there are infinitely many primes of the form 4k + 3.

A: We have to show that there are  infinitely many primes of the form 4k+3.

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

A:

Q: For sufficiently lazpe n en ns n Zus (en n)

A: According to this question, we know that

Q: Show that n!> n3 if n is large enough.

A:

Q: 3. Show that the weighted arithmetic mean of first n natural numbers whose weights are equal to the…

A:

Q: 8. How many positive integers less than 1000 a) have exactly three decimal digits? b) have an odd…

A: According to our guideliness we can answer only the first 3 subparts. Positive integers less than…

Q: The formula for calculating the sum of all natural integers from 1 to n is well-known: n² + n 2 Sn =…

A: Given:  Sn= 1+2+3+.......................+n = n2+n2and,Qn=1.1+2.2+3.3+...............+n.n=n33+n22+n6

Q: Suppose we have L-shaped tiles made up of three squares: Prove that for all n > 1, it's possible to…

A: Suppose we have L shaped tiles Prove that for all n>=1 it's possible to lay these tiles down to…

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

A:

Q: 5. Prove that the sum of the first n natural numbers is equal to n(n+ 1)/2, and deduce that the sum…

A: Sum of n number is given by

Q: The formula for calculating the sum of all natural integers from 1 to n is well-known: n² Sn…

A: This question is related to principal of Mathematical induction . solve in step 2.

Q: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

A: let n=p1a1p2a2....pmam be the prime factorization of n  Here p1,p2....pm are all distinct primes and…

Q: What is the largest number in the row starting "1 4 ..." of Pascal's triangle?   What is the sum…

A: To find what is the largest number in the row starting "1 4 ..." of Pascal's triangle? To find what…

Q: (3) Find the formula for the number of ways for putting n identical balls into k distinct boxes so…

A:

Q: Prove that the number of partitions of n into parts of size 1 and 2 is equal to the number of…

A:

Q: 1.How many license plates consisting of 6 digits (0 to 9) have their digits in strictly increasing…

A: Given that, How many license plates consisting of 6 digits (0 to 9) have their digits in strictly…

Q: Compute the sum of all integers between 60 and 150 that are exactly divisible by 8.

A: Consider the given range of number 60 to 150

Q: 1. How many positive integers of 5-digit-number can be formed from the prime numbers starting at 1…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

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: Discrete

A:

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

A:

Q: 32. Prove that 3 divides n + 2n whenever n is a positive integer.

A:

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: 7. The average value of thirteen different natural numbers was 20. If P was the highest number among…

A: In this question we have to find the minimum value of P.

Q: Show that there are infinitely many primes of the form 6k + 5.

A:

Q: 2. A group contains k men and k women, where k is a positive integer. How many ways are there to…

A:

Q: Show that there are infinitely many prime numbers.  0 /∈ N. (zero is not an element of natural…

A:

Q: 6. Each of the three vertices of a triangle is connected by n line segments to n distinct points on…

A: Given that, Each of the three vertices of a triangle is connected by n line segments to n distinct…

Q: 27. Determine the number of n-digit numbers with all digits odd, such that 1 and 3 each occur a…

A:

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

A:

Q: Show that the weighted arithmetic mean of first 'n' natural numbers whose weight are equal to the…

A:

Q: Suppose that an urn initially contains r red and b blue balls. At each stage a ball is randomly…

A: Given, number of red balls initially in urn = r number of blue balls initially in urn = b At the…

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: Show that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.

A:

Q: 6. (a) Show, using the pigeonhole principle, that in any set of 5 integers, at least two have the…

A: (a) Show that any set of 5 integers at least have two the same remainder when it is divided by 4…

Q: 8. Show that, at the n stage of the construction of the Cantor Middle- Thirds set, the sum of the…

A: Cantor Middle-thirds set: It is created by repeatedly deleting the open middle thirds of a set of a…

Q: 5. Invent a positive integer a that has at least 8 decimal digits and a ones digit of 7. Estimate…

A:

Q: A. The number of permutations of length n that avoid 132 and have the additional property that their…

A: Based on the additional property, the first entry is larger than the second entry is Cn-1, (the…

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: Suppose that there are n = 2k teams in an eliminationtournament, where there are n∕2 games in the…

A:

Q: Assuming every integer is either even or odd, How would I show that for any integer, the difference…

A: Assume that every integer is either even or odd. Let the integer considered say x is even. Then by…

Q: 1. How many combinations are there to pick r objects from 2n objects numbered from 1 to 2n with n >…

A: For the solution of the problem follow the next step.

Q: ind the number of ways to choose k children from a group of n children arranged in a circle so that…

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

Q: Prove that if n > 1 is a number with φ(n) = n/k, then k = 2,3.

A:

Q: (a) Find two consecutive primes that differ by at least 10. (b) Prove that there are arbitrarily…

A: According to the given information, for part (a) it is required to find the two find the two…

Q: Suppose that n students take a quiz and their scores sum to 150. What is the smallest number n to…

A: Pigeonhole Principle: The pigeonhole principle states that if n items are put into m containers with…