Exercise 3. Give a combinatoric argument to prove the identity 2 2n (3.1) j=0 Hint: A subset of n persons is selected from 2n persons, n female and n male. Count the number of n person subsets in two different ways.

Q: c) Prove that the following code returns z" by structural induction func power (x,y): if (y 0):…

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Q: 3. Show by mathematical induction that, for all n2 0, the equality +++.+ = 2 – +2 holds. ... 2Th

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Q: * For each integer n, let Cn = {n²,n³, nª}. How many elements are in each of C-2, C-1, Co, and C3?…

A: The set always contains the distinct elements. Given Cn = n2 , n3 , n4 Consider C-2 = (-2)2 , (-2)3…

Q: 1. Show that the principle of mathematical Induction implies that I is the Ieast positive natural…

A: To find gcd(493,899)

Q: H2) Prove, using mathematical induction, that 32N-1 + 1 is divisible by 4 for any NE N. (Hint: A…

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Q: 100 Σ (3* + 2k - 1) k=1

A: ∑k=11003k+2k-1

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: Using the generating function to determine the number of ways  18 identical balloons can be given…

A: Given that 18 identical balloons can be given to 3 children such that each child receives at least 5…

Q: Show that 1+2+3+… +n = (";").

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Q: 10. Use induction to show that 1x2 + 2x3 + + n(n+1) = " for all n € N.

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Q: 9. Using a proof by induction, prove that for every natural number n the quantity 11n -6 is…

A: The statement is not true. Tale n=4

Q: Let a = (17 3)(5 94 2) and B = (2 3)(7 4)(5 1 8) be elements of S10. Then, a²B- is O (1 4)(27 539 8)…

A: Given permutations: α=(1 7 3)(5 9 4 2), β=(2 3)(7 4)(5 1 8). To find: α2β-1.

Q: 4. Find the union and intersection of these families, and prove your answers. (a) A = {A, ne N},…

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Q: Prove C(n,0)=C(n,n)=1 for all whole numbers n

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Q: Use mathematical induction to prove Theorem 1 in Sec-

A: To prove that if b is an integer, where b>1, and n is a positive integer, then n can be expressed…

Q: Use the permutation and combination formulas to prove the following. +k – 1) (n + k – 1 (b). V k, n…

A: We have to  prove given formula using peemutation and combination

Q: 5. Use the principle of mathematical induction to complete the proof that the number of rays…

A: Kindly refer to Step 2 for the solution.

Q: Give a combinatorial proof the following statement 1(n−0) + 2(n−1) + 3(n−2) + 4(n−3) +···+ (n−1)2 +…

A: To give a combinatorial proof for an identity, we do the following steps Find a counting problem…

Q: Suppose that n is an integer such that n≥3. Using a combinatorial proof, show that:

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Q: 4. a) 10 samples of different publishers are on display at a book store customers.…

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Q: Find all n € N such that 2n +1 < 2ª. Prove your answer by induction. Find all n E N such that n² <…

A: First we find all the natural number n for which the given expression is true. Then we prove the…

Q: (5) Suppose thath and k are positive integers and that n = hk. Write a combinatorial proof that (h!)…

A: Suppose h and k are positive integers and that n=hk. Claim: n!h!k is an integer. Proof: Consider the…

Q: 5) Let a, = 2a +5,a, =1, use induction to show that a = 6(2")-5 for all n20.

A: Given that an=2an-1+5 and a0=1. We have to show that an=62n-5 for all n≥0. We have to use the…

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

A: Given information: The following identity is provided: n02+n12+n22+................+nn2=2nn It is…

Q: Show that if h+1 distinct integers are, Chasen from the Set El,2,en.,4n}. then there are always two…

A: To show that choosing n+1 distinct integers from the set {1, 2, 3, ..., 4n} then there are always…

Q: Roblem (6) Drcuss the unik Ganv (a) n+) (). an n+2 nal

A: (a) Given that ∑n=1∞1-1n+2xn+1 on (0,1). Take an=1-1n+2 The formula to find the convergence is,…

Q: A tennis tournament has n = 64 players, who are drawn uniformly at random to play against each other…

A: Given that n = 64

Q: Q1. Let x and y be two integers. Prove that for all integers m 2 1, x™ – y™ is a multiple of x-y…

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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:

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

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Q: prove by induction: 1+4+7++(3n-2)3水3-) for n71

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Q: (16) Prove that 22n – 1 is divisible by 3 for all n € N.

A: