Prove that the number of partitions of n into at most k parts is equal to that of partitions of n + k into exactly k parts.

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: How many elements of order 4 does A has.

A: To find Number of elements of order 4 does A6 has

Q: Show that the sum of all  odd  integers between 1 and 1000 which are divisible by 3 is 83667

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Q: Faces of a polyhedron are painted into two colors, so that no two faces sharing an edge have the…

A: Consider the provided question, We need to prove that the number of edges of all faces of polyhedron…

Q: Assume an unlimited supply of coins worth 5 cents and 9 cents. Prove that some

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Q: Given any whole number w, how many whole number triple decompositions does w have? Prove it.

A: A number is said to be whole when it begins from 0 and has no end. Decomposition of number means…

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

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Q: If the interval [2,7] is partitioned into n= 8 subintervals of equal length, what is Ax? Ax = |…

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Q: Show that the number of integer partitions (of any size) with at most n parts and with largest part…

A: Let Pm,n;N denote the number of partitions of N with at most n parts, each of size at most and with…

Q: find a way to partition N into not two but infinitely many subsets with each subset containing…

A: All the natural numbers can be written in prime factorization form. The natural numbers are the set…

Q: If the interval [4,9] is partitioned into n = 5 subintervals of equal length, what is Ax? Ax = (Type…

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Q: Prove that the summation of three odd integers is odd.

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Q: Show that for any positive integer n, with n not equal to 2, there exists a partition of n that is…

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Q: Prove that (n!)! is divisible by (n!)

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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: Find integers x and y such that 3220x + 6006y = gcd(3220,6006).

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Q: Is there a formula to find the number of partitions in a set with n-elements?

A: Bell numbers, Bn, gives the number of partitions of a set with n elements It counts the number of…

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

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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: 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: What is the conjugate of the partition (n – k) + k, where n 2 2 and n = isks \n/2]?

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

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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: Use Fermat's Theorem to show that n 13 - n is divisible by 2730 for all n.

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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: Show that the generating function for the number of self-conjugate partitions of n is (1– x²)(1 –…

A: To Show : Show that the generating function for the number of self-conjugate partitions of n is  ∑k…

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: Prove or disprove that all chessboards of these shapes can be completely covered using L-shaped…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 2. Prove that the Cartesian product N Nis countable

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Q: find the sum of all even intergers less than 1000 that are exactly divisible by 7

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Q: Show that the number of partitions of n in which odd parts are not repeated but even parts can occur…

A: Let n=4 Then number of partitions of n=4 in which odd parts are not repeated but even parts can…

Q: How many bitstrings are there of length 2n that contain exactly n “O"s and n “1"s? Prove your result

A: Given that,   The length of bitstring =2n, i.e If we have a bitstring, it should only contain 0's…

Q: Find the number of self-conjugate partitions of 17 and list all of them.

A: There are 3 odd distinct partitions in 17. 13+3+111+5+19+5+3

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: how many positive intergers less than 1150 are multiples of 8, 5, 3. using the principles of…

A: Floor Function: The floor function , also called the greatest integer function or integer value,…

Q: Prove that for every positive integer n the number (1 – /2)" is not ratioanal.

A: Given: The given number is, (1-2)n where n is the positive integer.   To prove: The number (1-2)n is…

Q: How many zero divisors are in Z,, where n = 32547?

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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 numbers of the form k/5", where k is an integer and n a positive integer, is…

A: The set of numbers of the form k5n where k is an integer are dense if every real number has rational…

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

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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: How many 5-tuples (x1, x2, x3, x4, X5) of positive integers from 1 to n are there? How many 5-tuples…

A: Since you have asked multiple questions, we will solve the Question Q1 (a) and (b) and (c) for you.…

Q: b) If a collection of objects consists of r, identical objects of type I, r, identical objects of…

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Q: Prove that the number of partitions of n into exactly k parts is equal to the number of partitions…

A: Each partitioning can be drawn out in the following manner: draw a column of xs that represents the…

Q: If the interval [3,10] is partitioned into n = 8 subintervals of equal length, what is Ax? Ax =…

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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: As we know that any truples h,i,j,k,m with n≥h≥i≥j≥k≥m≥1  can be represented as a string of n-1…

Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…