Suppose a, n E Z and 0 < a< n. Prove that ()- (":")- (:-) n - 1 %3D + a a а- 1 in two different ways: firstly using the formula for (") ering subsets of {1,..., n}. in terms of factorials; and secondly by consid-

Q: 2. A derangement is a permutation TE Sn with no fixed points, i.e. none of the elements remain in…

A: Introduction: Derangement is a very significant in combinatorics. Derangement is the process where…

Q: Compute f2,f3, and f-1 for each of the following permutations.   f= (1,5,2,4)

A: Given: f=(1,5,2,4) To determine the f2 f3 f-1

Q: A box contains five cards with even numbers and seven cards with odd numbers. If two cards are drawn…

A: Given : n (even) = 5 n (odd ) = 7 n (total) = 5 + 7               = 12 number of cards drawn with…

Q: Let S be the subset of (0, 1) consisting of numbers whose digits (in the decimal expansion)…

A: I have just used a property of countable set

Q: Let n E N. Prove by mathematical induction that 3* = 4". k k=0

A: We have to prove given expression using mathematical induction.

Q: By a result of Landau (1953), we know that every tournament has a king (a vertex from which every…

A: It is given that By a result of Landau (1953), we know that every tournament has a king (a vertex…

Q: 1. A restaurant is preparing a large round table with 2 +1 seats (the seats are unla- belled). The…

A: Given, Number of available seats = 2n+1 Number of red napkins = n+1  Number of blue napkins = n

Q: This exercise uses one or more of the following results from Section 5.2. Exercise 5.2.10: For every…

A: We will use the given information to complete the incomplete proof of given statement.

Q: п 12. Show that Ë O)=)2"-". Using Combinatorial Proof. k=0

A: To prove: ∑k=0nnkkm=nm2n-m

Q: Find f (n) when n = 4k, where f satisfies the recurrencerelation f (n) = 5f (n∕4) + 6n, with f (1) =…

A:

Q: Suppose a, n E Z and 0 < a<n. Prove that )-(".")-(C ) n- 1 in = a a а — 1 in two different ways:…

A: We have to prove, na = n-1a+n-1a-1 First let us prove this using formula. Using Formula…

Q: THEOREM 2.4 The number of distinct permutations of n things of which n1 are of one kind, n2 of a…

A: Given: Theorem 2.4 and Theorem 2.5

Q: Prove that k() = n(1) (n = 1) for integers n and k with 1 ≤ k ≤ n, using a: a) combinatorial proof;

A:

Q: Find the order of the residue class 5 in Z/29Z.

A:

Q: Prove by induction that (2n+1)3 is an odd number for all positiveintegers n

A:

Q: Find a Big-O estimate of (2n+n2)(n3+3n)

A: Given the given expression is 2n+n2n3+3n. Big-O notation: the fn is Ogn for positive constant C and…

Q: Derive a formula for E-1 Qiwhere Qi represents number of edges of n-dimensional cube and prove the…

A: As indicated, we shall use mathematical induction.

Q: Find the smallest positive integer n, пп such that 0 and 0 are 2 coterminal. And give an example

A: Coterminal angles are the angles in the standard position having a common terminal sides.

Q: Prove that () = ) + (") for every o sk sn. %3D Hint: (for Combinatorial proof) Consider, In the…

A:

Q: Show that if A is skew symmetric, then Ak is skew symmetric for any positive odd integer k.

A: Given that A is skew symmetric matrix. We need to prove Ak is a skew symmetric for any positive odd…

Q: 3. Suppose a, nEZ and 0 < a< n. Prove that - u 1 + a a (:) in terms of factorials; and secondly by…

A:

Q: السؤال 2 For all n E N, n> 1, there exists a prime p such that p|n. ylan O İhi O

A: True

Q: /2.11 The resumés of two male applicants for a college teaching position in chemistry are placed in…

A:

Q: Assume that n > 2 and x E A is a fixed element. How many unordered selections without repeats of k…

A: N&gt;2, X is the element A is a fixed element  How many unordered selections withouts repeats of  k…

Q: Consider the following theorem. Theorem 9.5.1: The number of subsets of size r that can be chosen…

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: Let p be prime. Prove that if H ≤ G, |G| = 2p and H is not normal, then |H| = 2.

A:

Q: 3.(d) Let a, be the number of edges of W, and b, be the number of edges of C, Find a recurrence…

A: Solution of part (d): Given that an is the number of edges of Wn and bn is the number of edges of…

Q: Prove that there exists an n ∈ N such that 7 | (3^n − 1).

A:

Q: Theorem 24. If ɛ > 0, then there is an integer N EN such that 0 < + < e. .

A:

Q: 6. Verify Euler's Theorem for n = 16 and a = 5. (7 from 6.5)

A:

Q: 4. If G = (Z8,) then G has an element of order 5. *True *False

A:

Q: Suppose that P(A;) = 1 for all i = 1, 2, .... Show that P(N A;) = 1.

A:

Q: Prove that Fisher's ideal index number lies between Laspeyre's and Paasche's index number

A:

Q: 3. Use the permutation and combination formulas to prove the following. (a). For all integers n 2 3,…

A: Given n≥3, n€ Z (integer ) We have to prove that P(n, 3)+2P(n, 2)=nP(n, 2)

Q: Show that if M is a number less than 1, then the terms of {n/(n + 1)} eventually exceed M. That is,…

A:

Q: Prove that ℤ/n has exactly n elements.

A:

Q: 2. Show that for every n 2 2 and every 0 <k<n- 2, within the triangle of Binomial Coefficients k+2…

A: We will use properties of binomial coefficient  nCr  +  nCr-1=n+1Cr+1Note : nCr=nr=n!r! n-r!   Proof…

Q: Suppose n > 3 is an integer. Prove that in Sn every even permutation is a product of cycles of…

A:

Q: 5. Prove that for all n EN sufficiently large, 3" > 2n² + 1.

A:

Q: (i) Prove that there are infinitely many primes of the form p = 6n – 1. -

A: Given:    Infinitely many primes of the form p=6n-1     Proof:         Suppose p1, . . . , pn are…

Q: 7. Prove that a permutation in S. n ≥ 2, cannot be both even and odd.

A:

Q: Obtain the generating function for the number of k-part compositions of n in which the ith part does…

A: Solution :-

Q: 16. Prove that, for integers a and b, gcd(a, a + b) gcd(a, b). [Hint: Mimic the proof of Lemma…

A:

Q: Use the permutation and combination formulas to prove the following (a). For all integers n > k > 2,…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 2. Suppose that P(A;) 1 for all i = 1, 2, .... Show that P(N A¡) = 1.

A: Given problem Given that

Q: Suppose a, b ∈ Zare relatively prime, and c|a. Prove gcd(b, c) = 1.

A: Handwriting solution.

Q: (This exercise constructs another proof of the infinitude of pimes) Show that the integer Qn = n! +…

A:

Q: 3. For every integer k > 0, define A = {2*n : neN and n is odd}. Prove that {Ag : k = 0, 1, 2, 3,…

A: