Find the number of objects for the following: -the number of bijections from [n] to [n] such that there are exactly 3 elements of [n] which are NOT fixed points

Q: Find the value of k such that A is singular. 1 0 5 3 -1 0 2 k k =

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Q: Prove that for all positive integers k and n, with k < n, n-1) k-1/ k 1 k ... k k- 1 k- 1

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Q: Let X = {x E Z|x = Y = {y € Z]y = 3n for some integer n}. 6m for some integer m} and let %D a. Prove…

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Q: 5**) (a) Prove that for all positive real numbers n, r, and s, if rs < n, then r < Vn or s < Vn.

A: As per Bartleby's answering policy, we can answer only one question, so kindly post the next…

Q: is an element in X that is not in Y. O No element in Y is in X. O There is an element in Y that is…

A: We will determine the meaning of subset and solve the problem.

Q: 3. Prove that for all x > 0 and all positive integers n, x3 et >1+x + 2! 3! n!

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Q: How many k-lists {r1, x2, ..., xk} are possible, such that each x; is a positive integer and 1 <x1 <…

A: Here each xi is a positive integer and 1≤x1≤x2≤...≤xk≤n.

Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: Prove that E, k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

A: Note that there is a typo in the question. We need to show that,                            We will…

Q: Prove that for all n E N, 1 1x 2 x 3 n(n + 3) 4(n + 1){n + 2 1 1 2 x 3 x 4 n(n + 1)(n+2)

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Q: Find an integer n such that U(n) is isomorphic to Z2 ⊕ Z4 ⊕ Z9

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Q: Find the value of k such that A is singular. 1 0 3 4 -1 0 2 4 k k =

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Q: Find a closed formula for a, if ak = 5n² + 4n k=1 an =

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Q: a) Prove that, for all integers K > 5 Р(к +1,5) — Р(К,5) %3D 5P (К, 4)

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Q: n Σ k (k!) = (n+ 1)! - 1. %3D k=1

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Q: -), prove that for all n E N: (a) n-1 E(:)-E- n k k=1 k=0 (b) ¿(:)-- = 2" k k=0 (c) El-1)* E-1» (:)…

A: According to Bartleby policy we solve up to 3-subparts. Please repost #d and mention it.   b)…

Q: Find a positive integer m > 3 such that there are at least 3 integers n in Zm such that n2 = 1.

A: We have to find a positive integer m≥3 such that there are at least 3 integer n in Zm such that n2=1…

Q: with Zeros of 2i with integer coeRficients that haf E Puno

A: We need to find the polynomial of degree three with integer coefficient

Q: 11. Prove that if d divides a, then dk divides ak for every positive integer k.

A: What is Divisibility: Divisibility is the base of number theory. For two given integer with one…

Q: Find the supremum and infimum of the set A := ∩ ∞ n=1 (1 −1/n ,1 +1/n ). And prove it

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Q: Prove that for all positive integers k and n, with k < n, (:) - () - (^) n k 1 k n 1 k k – 1 k – 1 k…

A: 1+kk-1+K+1k-1+   ... ...+n-1k-1 add and subtract 1 and write 1 as a combination of k objects taking…

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

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Q: Find the missing parts of the Wronskian: x Inx ? W = inx - and 4 O x2Inx and x3Inx O 1 and Inx O 1…

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

A: We will solve the problem

Q: Compute the infimum and supremum of the set {n+(1/n)" : n E N} if they exist.

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Q: Consider the following theorem. (") and is given by the formula Theorem 9.5.1: The number of subsets…

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Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: Prove that for all positive integers k < n, the equality n n n =2n-k k k i=k holds.

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

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Q: Let r(n) be the number of n-permutations whose square is the identity permutation. Prove that if n >…

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Q: Find the elements of Z24 of order 8. (Hint: o(1)=24)

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Q: Is the following statement true or false? For every odd integer n, To answer this question, let n be…

A: Let, n be any odd integer. By definition of odd, n=2k+12, for some integer k So, by solution,…

Q: Find the smallest integer in the given set. {xEZx> 0 and x = 4s + 6t for some s, t in Z} Select one:…

A: Given: To explain the correct option of the given statement as follows,

Q: Prove that gn = 5 · 3n + 7 · 2n for all integers n ≥ 0.

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Q: п Find a formula for II (1- ) : and verify. k2 k=2

A: We will expand the terms and check which terms are getting cancel each other

Q: Let A = {4n^2 + 1 | n ∈ Z}. Show that the size of A is equal to the size of Z≥0. Z≥0 is the set of…

A: A set A is said to be countable  if there is a bijective map between A and set of natural numbers .…

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

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Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZ≈ Z/mnZ if and only if m and n are…

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Q: even and n > 4, show that rk is an element of order 2 which commutes with all elements of D2n

A: Given, n = 2k is even and n ≥ 4, We have to show that rk is an element of order 2 which commutes…

Q: Prove that, for all integers K and L, there is at least one pair of integers (a, b) for which K2 +…

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

A: Solution :-

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: Find the value of k such that A is singular. 1 0 6 4 -1 0 6 2 k

A: Given. A is a 3X3 matrix. We need to find the value of k for which the matrix is singular. Singular…

Q: Find the value of k such that A is singular. 0 4 6 -1 0 4 2 k k =

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Q: 4. Verify thatf (x) = x³ – x2 – 20x + 6 satisfies the three conditions of Rolle's Theorem on [0,5].…

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Q: a) Prove, for all integers k, that 2k-1 is odd. b)prove, for all integers x, if x is odd, then x =…

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Q: 2. List all the elements of (Z30, Ð) that are of order 15.

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Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.