(1234 56 314562, Let o = find o5 (1234 56 314562, (1234 56) 612435, O None of them 1234 56) (124365 1234 56 435621

Q: 3. How many 4-digit positive integers are there that are even or contain no O's?

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Q: Prove that the hundreds digit of n20 is even.

A: To Prove that the hundreds digit of n20 is even.

Q: Let a be an integer greater than 0 and b be an integer greater than or equal to 0. Let a be greater…

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Q: ind witnesses to show that n^3 − 5n^2 − 4n is O(n^3).

A: In this question, order of polynomial can be found with the help of Big-O notation. Big-O Notation…

Q: (D) Let s be an integer such that 1 ss < 104651 and 24578° mod 104651 = 3190 Find s and explain how…

A: Let s be the integer such that and Purpose: To find the value of s

Q: E. Prove by Mathematical Induction that for any natural number n 1+5+9+13 + .. .+ (4n – 3) = n(2n –…

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Q: Prove that by mathematically Induction 2n < (n + 1)!, for all integers n ≥ 2.

A: Let s(n):2n&lt;(n+1)!,n≥2...(1) Now, Substitute n=1 in (1), so, 2(2)≤(2+1)!⇒4≤3!⇒4&lt;6 Which is…

Q: (123456 Let o = find o25 314562 ect one: 123456 612 435) (123456 124365 (123456 314562 (123456…

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Q: Let 812 c3 8°c2 S = 1+ 8°c + 2! 816c4 3! 4! If In S = 131072, then find c. Türkçe: Eğer 812 c3 + 2!…

A: We have to find c

Q: 4. Let A = of O 2| Calculate

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Q: (b) Let Ho =1, H1 =3, H2 =9, and define Hk = Hk-1 + 3Hk-2 + 3Hk-3 for k23. Show by induction that Hk…

A: We will prove the result by induction  H0=1≤30 H1=3≤31 H2=9≤32 The result is true for k=0,1,2 Assume…

Q: Prove, by induction, that, for n2 5, the nth Fibonacci number satisfies Fn s 2".

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Q: Use a proof by contradiction to prome that integer ond 3n+z then

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Q: Suppose a set of 8 numbers are selected from the set: {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13;…

A: It is given that 8 numbers are selected from the set 1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: Prove the following claims: (a) If n is an integer between 0 and 10, then n = a² +b² +c² +d², for…

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Q: Let n eN 1. Show that o (n²) = ng(n) 2. Show that o (n) +

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Q: 1. Let n E N. Prove that n and nº have the same last digit.

A: See the attachment

Q: Find the following Wronskian: w(2, 31, —е 4, te 4) O 8 O 1536e8! O -8 O - 1536e8: O 1536e¬8 -8r O -…

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Q: Let 88c² 8123 + 816c4 S = 1 +8¹c+ 2! 3! 4! If In S 131072, then find c. Türkçe: Eğer 88c² 8123 816c4…

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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: Please show steps

A: Consider X is a random variable defined as left-handed individual.And p is the probability of…

Q: How many 2-permutations are there of a set of eight objects? Prove that for all integers n > 2, P(n…

A: The formula of permutations is given as nPr = P(n, r) = n!/(n-r)!

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: Evaluate E=z*p1•q1n1+p2•q2n2 for z*=2.33​, n1=2914​, n2=7807​, p1=0.0124​, q1=0.9876​, p2=0.0014​,…

A: E=(2.33×0.0124)·0.9876×2914+0.0014×0.9986×7807=0.028892×2877.8664+10.914498=83.147316+10.914498=94.0…

Q: prove the following entities: or all integers n>= 1, * n(n + 1)(n + 2)(n +3) 1.2.3+2.3-4+ .+ n(n+…

A: We will use principle of mathematical induction .

Q: Prove that 36n is not O(35n)

A: Let, 36n∈O35n

Q: Show that 2n is O(3n) but not that 3n is not O(2n)

A: We need to show that 2n is O3n but 3n is not O2n Definition of Big O notation: Let f be a real…

Q: q2-  91.2% of 1980 is: Select one: 18057.6 180576 180.576 1805.76 18.0576

A: We are asking to find 91.2% 0f 1980

Q: Let X-N(65,36) P(x<70)= O 0.50798 none O 0.98077 O 0.76973

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: Use mathmatical induction to prove that for all intergers n is greater than or equal to 1.…

A: For n = 1,       LHS = 4 [only 1 term]      RHS = 2.12 + 2.1 = 2.1 + 2.1 = 4 Hence, LHS = RHS.  So,…

Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: Prove by mathematical induction that Vn E Nwith n 2 4 2 n3

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Q: Prove by the method of Mathematical Induction that, E 2' = 2" – 1 for any positive integer n.

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Q: Let a1 = a2 = 1 and suppose an = an-1 + an-2 for n ≥ 3. Prove that a1a2 + a2a3 + · · · + a2n-1a2n =…

A: We have given that , a1 = a2 = 1 Also , an = an-1 + an-2 We need to prove that ,  a1a2 + a2a3 +…

Q: Let a1, a2, az, ... be the sequence of numbers defined by a1 = 1, an = V2+ an-1 if n > 2. Prove by…

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Q: Let an = n1.9 + n².!. Prove or disprove that ɑn = O(n²).

A: Given: an=n1.9+n2.1 ,To be prove an=On2

Q: 4. Find x in ASTU. * 17 15 O 2 O 8 O v32 O V514

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Q: .3 ). Show that x is O(x*) but that x* is not O(x³).

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Q: 11. Let n = 7820972339. Suppose %3D $(n) = 7820794224. %3D Find p and q.

A: n=7820972339 ϕ(n)=782074224 To find p and q.

Q: =) Prove that (*) + ("*) =n“, where n is a positive integer. n", where n is a positive integer. 2

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Q: Let n be a positive integer. Which among the following statements is true? * O n+1 and n+ 2 are not…

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Q: b) For which positive integer n is it true that 2"> n³? c) Prove your answer in (b) above using…

A: b. 2n&gt;n3 Now for n=1 21&gt;132&gt;1 For n=2 22&gt;234≯8 For n=3 23&gt;338≯27

Q: Show that sup{ 1- n: n EN} = 1.

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Q: E. Prove by Mathematical Induction that for any natural number n. 1+5+9+13 + .. + (4n – 3) = n(2n –…

A: The mathematical induction can be used to prove the statements. The proof by mathematical induction…

Q: 9:Let S [o,1] Prove that inf s=o

A:

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.