Let k, n be positive integers with k < n. Give a combinatorial proof of the identity (in the form given) () - (",") - (:-)- n-3 n- n - 3 %3D k - 1 k - 1

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

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Q: Show that 1+2+3+… +n = (";").

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Q: Find the smallest positive integer in the given set. (a) {6u + 15v : u, v ∈Z}    (b) {12r + 17s : r,…

A: Given: (a) { 6u+15v:u,v∈Z }(b) { 12r+17s:r,s∈Z }

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Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

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Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: a)Show by one combinatorial method that: 2 2 n C) - () - (") 2n +... + 2 n

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Q: Find the general solution of the recursive relation an ○ an = a (1 + √√14)” + ß(1 − √√14)” an = a(1…

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Q: Using mathematical induction, prove that this equation holds for any n 2 1: n(n + 1)(n + 2) 1x 2+ 2…

A: Follow the process below and use induction first replace n with k and then n with k+1.

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: give a combinatorial proof of the identity  P(n,k) * (n-kCm-k) = (nCm) * P(m,k) where all…

A: Detailed proof is provided in the next step.

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Q: Question: Evaluate each following expressions a.) If x =0   evaluate 4x**2 + 5(9 +x) + x **5 b.)…

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Q: For each integer n2 2, let P(n) be the following equation (n – 1)n(2n + 5) (i – 1)(i + 1) = 72 - 6…

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Q: A) Express the following using favtorials and simplify: ([n] [2]). B) Show the following using…

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Q: . Use mathematical induction to show that n Σ r=1 (2r − 1)² = 1² + 3² +5² + + (2n − 1)² = = n(2n −…

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Q: Find the general solution of the recursive relation an = 2 an-1 - 6 an-2 for every integer n > 1. a,…

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Q: Which of the following is true? Select one: a. (n+1)= n(n) for any real number b. г(n) = n(n+1) for…

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Q: Use combinatorial proof to show that 10 (2n - 5 3 n - 2 n | %3D 10 k 10 – k

A: Given:

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Q: Give a combinatorial proof for the identity 1+2+3+..+n= (",").

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Q: ite P(0). Is P(0) true? Write P(k). Write P(k + 1). In a proof by mathematical induction that this…

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Q: Prove:  C(n,k) = C(n-1,k-1)+C(n-1,k) for natural numbers k with 0<k<n.

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Q: Write the recursive definition for an = pn n! where a1 = p.

A: The recursive formula is used to describe the relationship between two terms of any sequence. The…