Problem 3. For any integers n, k,r where n > k > r > 0, give a combinatorial proof of the following identity. ()()-OC) n - r %3D k -
Q: Problem 2. Find all triples of integers (x, y, z) that satisfy r² + 2y? = z2.
A: (x,y,z)=(k|b2−2a2|,k(2ab),k(b2+2a2)),a,b,k∈Z+
Q: Problem 1. Give an alternative proof for Gauss;s Theorem by using that O is multiplicative.
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Q: Problem 4 For all n E Z, let An = {n – 3, n, n +3}. Identify the set Unen An. (Hint/Warning: the…
A: Given: For all n∈ℤ, An=n-3, n, n+3
Q: Problem 5. Using the definitions for O and N, prove formally that a) n³+15n+2 is O(nª) b) 2n³ +25n…
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Q: Problem 6.4 Suppose that n is a positive integer. Justify the identity 2 2n ()
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Q: Problem#1: Determine the number of integers among the first 100,000 positive integers that contain…
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Q: Problem 3.1 Prove that if x is a positive real number, then LV[]] = [V]
A:
Q: Problem 2 Comparing two means Consider two measuring instruments that are used to measure the…
A: Note: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question…
Q: Problem 4. Prove by induction on the complexity of formula that A[x := x] is always defined and…
A:
Q: 7. Repeat problem 4-6 with s = 1.5° F instead of o = 1.5° F.
A: The sample mean , x = 98 sample standard deviation, s = 1.5 number of samples, n = 9 level of…
Q: Problem 6:- I toss a coin repeatedly until I observe the first tails at which point I stop. Let X be…
A:
Q: Problem 1: Show that Vn E N+ that gcd (fn, fn+1) = 1 where fn is the n-th Fibonacci number.
A: The provided information is The gcd(fn,fn+1)=1where fn is the number of n-th Fibonacci…
Q: Problem 2. MAGA hats are on sale at 90% off, so Don has purchased 19 of them. Now he wants to gift…
A: Multinational theorem provides an easy way to expands the powers of a sum of variables In above…
Q: Problem 3: Let n be an even positive integer. Show that when n people stand in a yard at mutually…
A: Yes
Q: Problem#1: Consider E = {a, b} a. L2 = {w over E||w| > 2 and |w| 2 and |w| < 5 and |w| is even }.…
A: As per bartleby guidelines for more than three subparts in a question only first three should be…
Q: Problem 7. Let f(x) = xe". Compute y', y", y", y(4), y(5). Determine a formula for ym) for any…
A:
Q: PROBLEM 3 Let h 2 1, z be a real number, and z> -1. Prove the following statement using mathe-…
A: CONCEPT: In order to prove that a statement is true for all natural number using Mathematical…
Q: Problem 4. Consider the two following subsets of the real numbers n 2n + 1 S = :n € N CR, T = EN CR.…
A: Supremum of a set is defined as, " If the set of all upper bounds of a set S has a smallest member…
Q: Determine the calculation result of Z1- Z2/Z3
A: a. (3-4)/4 = -1/4
Q: 1.6. Problem 6. Use Induction to prove that for all n E N21 E(1+ sin (i)) > 20 i=1
A: .
Q: Problem 3.1 Prove that if x is a positive real number, then LVL]] = LV]
A:
Q: Problem 5.2 Prove the hockeystick identity (**) 'n + k` 'n +r +1\ k k=0
A: We have to prove Hockeystick Identity by Pascal's Identity.
Q: Problem 3. Let a be an integer that is coprime with 240. Prove that 240|a4 – 1. -
A:
Q: Problem B2) Given the following figures made from gray squares: 3 4. n = 1 n = 2 n=44 a) Assuming…
A: (c) the number of grey squares used will be calculated as follows Each figure uses three…
Q: Problem 11.57db: Test the hypothesis Ho: B1 =0 against its alternative. Provide a p-value for the…
A: Let x be the lysine ingested, and y be the weight gain.
Q: problem 5 (1) State Euler’s Theorem (2) Verify Euler’s Theorem for n = 15 and a = 7.
A: Euler's Theorem:Suppose that n∈N, a∈Z and a and n are coprime.Then,aϕ(n)≡1 (mod n) we have…
Q: Problem 8. Let M(1) = E#(n). Suppose X > 1. Prove that M = 1.
A:
Q: Provide NEAT and COMPLETE solutions for the following problems. 4. Let n ∈ Z. Prove that if n is…
A: Given: Prove that if n is odd, then n = 4k + 1 or n = 4k + 3 for someinteger k.
Q: Find a general solution of each reducible second-order differ- ential equation in Problems 43-54.…
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Q: Problem 9.9 Show that o(35) = ¢(5) · ø(7), where ø(n) is Euler's totient function.
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Q: Problem 3: Let B, be the number of compositions of n in which the first part is odd. If n > 1, show…
A:
Q: For the extension field in problem 6, give the sum and product of these two elementsr+1, r² -2. You…
A: Solution
Q: Problem 7.3.5. Show that if (an)n=1 =D1 diverges to infinity then (a,)1 diverges. n=1
A: The given sequence ann=1∞ is diverges to infinity. That is limn→∞an=±∞. Without loss of generality,…
Q: Problem 7. Let F(n) = Edma>(d)oo(d). Give a formula for F(n). %3D
A: Given function is, Fn=∑dn,d≥1μdσ0d So when x=0 the function counts the positive divisors of n. Note…
Q: Solve the initial value problem %3D 2 — 4г, у(0) 3D —2. (Use symbolic notation and fractions where…
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Q: Problem 5 Evaluate L-1 B using the convolution theorem.
A:
Q: Problem 5.6. Let ze C. (a) Prove that |1*| is single-valued if and only if Im z = 0. (b) Find a…
A: This is the problem of complex analysis.
Q: Problem 1. Apply the interchanging order of summation technique to evaluate ΣΣ k j=1 k=j
A: We will solve this by using partial sum expression
Q: Problem 6) Verify the Chebyshev's rule: P(|X – µ| > ko) < for k = 2 and for X~Bin(25,0. 5).
A: See the hand written solution
Q: Problem 3. For real numbers x, we defined [z] as the unique integer such that [2] ≤ x < [x] +1.…
A: Let, x be a real number. Then the greatest integer function [x] as the unique integer, is defined as…
Q: Problem 1. Prove that if b>1, then for any K>0, there is n E N such that b" > K. (Hint: Use the…
A:
Q: 10 2i 5 + 20. lim 2 i=1 N n→" n
A:
Q: 7. Repeat problem 4-6 with s = 1.5° F instead of o = 1.5° F. %3D
A: From the given information, Sample size n =9 Sample mean, x̅=98 Sample standard deviation, s =1.5…
Q: Problem 2. For Example 7.7 in text (Page 335), show that (1). E(Ô,)= 0, 02 (2). Var(ê) п(п + 2)" n…
A:
Q: Problem 2. Suppose A E R"×n. Show that, for all x E R" xT Ax = x Agx, where A, is the symmetric part…
A:
Q: Problem #3: Consider the following statements. (1) Σ n=0 (-1)" 2n+1 X² (2n + 1)! = X- ∞ (iii) The…
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Q: Problem 7. Compute det A, where (a) det A = 0 (b) det A=-12 (c) det A = -24 (d) det A = 24 (e) None…
A: We Know that, Property A : If all the elements of a row (or column) of Matrix are zero, then the…
Q: Question 10 Expand g (x) = 10x-' in powers of (x – 1). a) O 10(-1)*-'(x – 1)* k=0 b) OE(-1)*-'(x –…
A:
Q: Problem 6: (i) Find the smallest prime that is of the form 709x² + 1061xy + 397y² for some x, y € Z.
A: Hi! Thank you for the question, As per the honor code, we are allowed to answer one question at a…
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- 30. Prove statement of Theorem : for all integers .Question # 4: Suppose that every student in a discrete mathematics classof 25 students is a freshman, a sophomore, or a junior. Show that there are at least nine freshmen, at least nine sophomores, or at least nine juniors in the class. Show that there are either at least three freshmen, atleast 19 sophomores, or at least five juniors in theclass.Question #07 i) A box contains three white balls, four black balls and three red balls. Find the number of ways in which three balls can be drawn from the box so that,a) At least one of the balls is black.b) With no restrictionc) One ball of each colors.ii) Using the digits 1, 2, 3 and 5, how many 4 digit numbers can be formed ifa) The first digit must be 1 and repetition of the digits is allowed?b) The first digit must be 1 and repetition of the digits is not allowed?c) The number must be divisible by 2 and repetion is allowed?b) The number must be divisible by 2 and repetion is not allowed?
- Note: In this problem set, the notation C(n, k) is used for the combination "n choose k." Alternate notation: ( ) For Question 11: A panel of 4 Democratic candidates, 4 Republican candidates, and 2 Independent candidates will participate in a political forum. 11.) If members of each party do not need to be grouped together, and we consider candidates from the same party (as well as Independents) as indistinguishable, in how many distinguishable ways can the candidates be seated in a row?Problem 1 For each of the following sets, if it is finite, state its exact cardinality; if it is infinite, state whether it is countable oruncountable and justify your answer:[a] The Cartesian product of the even integers with itself.[b] The negative rational numbers.[c] The real numbers between (-1 and 1).[d] Strings of all lengths comprised using the binary digits {0, 1}[e] H x H x H x H x H where H is the set of integers between 1 and 100.Prove that if n > 1 is a number with φ(n) = n/k, then k = 2,3.
- Question # 9: Suppose that a password for a computer system must haveat least 8, but no more than 12 characters, where eachcharacter in the password is a lowercase English letter,an uppercase English letter, a digit, or one of the six specialcharacters ∗, >, <, !, +, and =. a) How many different passwords are available for thiscomputer system? b) How many of these passwords contain at least one occurrenceof at least one of the six special characters?2. The physicist Frank Benford noticed in the 1930s that, for many sets of numbers in the real world, the leading digit* of the number is more likely to be a small number than a larger one (i.e., 1’s and 2’s are more common than 8’s and 9’s). In fact, many wide-ranging number sets obey “Benford’s Law,” which says that 1’s are about 30.1% of the leading digits, 2’s are 17.6%, 3’s are 12.5%, 4’s are 9.7%, 5’s are 7.9%, 6’s are 6.7%, 7’s are 5.8%, 8’s are 5.1%, and 9’s are 4.6%. Benford’s Law has been used to identify accounting fraud and made-up scientific data. *leading digit is the first significant digit; e.g., “1” in the number 132, “2” in the number .0026.S uppose I suspect my research collaborator of faking a dataset that represents the price that people say is the average amount they pay each month in health care deductibles and copayments, rounded to the nearest dollar.10 50 15 11 412 80 48 13 4250 10 68 41 790 15 139 147 1433 61 145 7 406 99 50 27 7513 40 3 21 1475 203 148 27…Rework problem 17 from section 1.4 of your text, involving a product code. Assume that product codes are formed from the letters Y, X, T, V, R, Z, and U, and consist of 5 not necessarily distinct letters arranged one after the other. For example, YYZUZ is a product code. (1) How many different product codes are there? (2) How many different product codes do not contain Y? (3) How many different product codes contain exactly one Z?
- find the solution to the recurrence relation t(n)=-t(n-1)+12 t(n-2) with base condition T(0)=2 and T(1)=3Question 3 Q20. Suppose a computer program has been initialized such that the following sets have been stored for use in any algorithm: A = {1, 2, 3, ..., 43}B = {-7, -6, -5, ..., 21} Consider the following algorithm, which represents one part of the whole computer program (comments may occur after the # symbol on any line and are not used in computations): #Part 1: computes A - B and its cardinality AminusB = set()for element in A: # this line runs through every element in A if not(element in B): #A - B is the set of elements that are in A and are not in B AminusB.add(element) # Add to AminusB every element in A if the element is also not in B n = len(AminusB) #len() returns the number of elements in the arrayprint(n) What value is printed as a result of executing this algorithm? Your Answer: Question 3 options: AnswerSuppose there is a machine that has an unlimited supply of eight different kinds of stickers. Each sticker is as likely to appear as another. Given this answer the following combinatorics problems: (i) How many stickers do you need to buy to make sure that you get three of the same sticker? (ii) How many ways can five different people each get a sticker, where there is at least one person who gets a lion sticker? Given that is matters which person gets which sticker (iii) How many ways can you get five stickers from the machine where every sticker is a different kind? Given that it does not matter which order the stickers come out
