How many permutations of [n] are possible in which no even numbers and no odd numbers are adjacent?

Q: 2. A derangement is a permutation TE Sn with no fixed points, i.e. none of the elements remain in…

A: Introduction: Derangement is a very significant in combinatorics. Derangement is the process where…

Q: Let n and m be two integers such that n> m > 0. If a bit string of length n is randomly selected,…

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Q: Prove by mathematical induction that for each natural number n 27, n! > 3"

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: (a) Show that if n 7 mod 8, then n + 6n 3 mod 8.

A: If, n = 7 (mod 8) So, n² = 7² (mod 8) So, n² = 49 (mod 8) So, n² = 1 (mod 8)

Q: Prove by the method of Mathematical Induction that, 3 divides n3 + 2n for any positive integer n.

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Q: Prove with induction the sum of first n Fibonacci numbers with even indices, i.e., f2 +f4 +···+f2n…

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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: Use ibn al-Haytham's procedure to derive the formula for the sum of the fifth powers of the…

A: We have to derive the formula 15+25+......+n5=n66+n52+5n412−n212 by using ibn al-Haytham's method…

Q: Prove by the method of Mathematical Induction that, 3 divides n + 2n for any positive integer n.

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Q: THEOREM 2.4 The number of distinct permutations of n things of which n1 are of one kind, n2 of a…

A: Given: Theorem 2.4 and Theorem 2.5

Q: 1. Prove by induction, for each n in N, 1+3+5+7+..... + (2n – 1) = n?

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Q: Use Mathematical Induction to prove that for all n eN with n > 2, 1 1 + + 2.3 1 +... + 1 n 1 - 1.2…

A: To prove using mathematical induction first we need to show that the given statement hold for…

Q: 1. Establish by mathematical induction that any postage of at least 12¢ can be obtained using 3¢ and…

A: For the solution follow the next steps.

Q: 8. Find the sum of all integers between 20 and 29, including these.

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Q: . Use mathematical induction to show that n" > 27 for all integers n > 2.

A: Given data:   The expression for the given inequality is nn≥2n.   The given inequality can be…

Q: Let o (n) be the sum of the divisors of n and let (n) be the Euler phi functions. Let n = = pq be…

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Q: Number theory.

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Q: prove (1+ )" in <4 for all n21

A: In the given question we have to prove that  1+1nn&lt;4∀n≥1 . we can prove this by two methods one…

Q: 6. Prove by contradiction: On the outside rim of a circular disk the integers from 1 through 30 are…

A: We need to prove the given statement.

Q: Prove the following by mathematical induction: п(n+ 1)(п+ 2) 1.2+2.3+3.4+……+n(n+1) = 3 for all…

A: We have to prove the given summation result using the principle of mathematical induction.

Q: 6. Prove that the product of an even permutation and an odd permutation is odd.

A: Let S be set on which permutations are defined. Let  α=(x1,x2,...,xk) and  σ=(y1,y2,...yn) be two…

Q: Which of the following permutations are odd? (823)(7546) none of these (1423)(456387)…

A: To check: If the options are odd or not

Q: 25. How does Cantor's theorem imply that there are infinitely-many different infinite cardi-…

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Q: Show that there are infinitely many primes of the form 6k + 5.

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Q: 6. Establish by mathematical induction that any postage of at least 12¢ can be obtained using 3¢ and…

A: 12¢=4×3¢+0×7¢13¢=2×3¢+1×7¢14¢=0×3¢+2×7¢15¢=5×3¢+0×7¢16¢=3×3¢+1×7¢17¢=1×3¢+2×7¢…

Q: Number of circular permutations of different things taken all at a time is (n-1)!. O a. True

A: Given statement :Number of circular permutations of different things taken all at a time is (n-1)!

Q: Let p be a prime number larger than 2. Suppose we uniformly sample for two times with replacement…

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Q: (b) Prove that there are infinitely many prime numbers of the form 6n + 5

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Q: prove that there exists a prime number p such that n <p< 2n for n = 10,11,12,13,14, 15, 16, 20.

A: for n=10 then take p=19, 10&lt;=p&lt;=20  for n=11 then take p=19, 11&lt;=p&lt;=22 for n=12 then…

Q: Prove by Mathematical Induction that: 1∙1! +2∙2!+ +...+n∙n!= (n + 1)! –1

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Q: 3. Use principle of mathematical induction to prove that (n)(n + 1) 1+ 2+3+ ·..+n = for n > 1.

A: Solution

Q: Use mathematical induction to show that n > 2n +1 for all integers n > 2.

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Q: 13.How many permutations are there of the digits in the string 12345 that leave 3 fixed but leave no…

A: Solution is given below:

Q: 2. Define sequences (an) and (bn) by letting an-1+ bn-1 an = bn = Van-1bn-1- Deduce from the…

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Q: 1) Use mathematical induction to show that any postage of at least 28 cents can be obtained using 5c…

A: Image is attached with detailed explanation.

Q: Show by mathematical induction, that for all n 2 1 n! > 2n-1

A: To show → by mathematical induction that                       V n ≥ I  , n I ≥ 2n-1Proof →Let ,…

Q: Use mathematical induction to prove that for all integers n ≥ 1, 6" 1 is divisible by 5.

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Q: Prove by induction that n! is O(n^n) for n ≥ 1

A: Definition of Big-O: f(n) ∈ O(g(n)) if f(n) ≤ C g(n), ∀n ≥ k, where C and k are positive constants.…

Q: 13. Determine the number of permutations of {1,2, . , 9} in which at least one odd integer is in its…

A: Solution : There are total 9 numbers from which five numbers  are odd (1, 3, 5, 7, 9) and four are…

Q: 7. Prove that a permutation in S. n ≥ 2, cannot be both even and odd.

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Q: Use mathematical induction to prove that for every integer n ≥ 2, if a set S has n elements, then…

A: To prove: For every integer n ≥ 2, if a set S has n elements, then the number of subsets of S with…

Q: 3. Use mathematical induction to prove that the sum of the cube of the first n positive integers…

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Q: 5. Suppose we have an enumeration of A ao, a1, A2, ·.. (1) without repetitions (i.e., all the a; are…

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Q: How many permutations of{1,2, . . . ,8}are there in which no even integer is a fixed point

A: We will permutations to find our required result.

Q: How many permutation g on {1,2,3,4} which satisfies g(1)=2 are possible? (Write the number you…

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Q: Prove: N × N is countable.

A: To prove ℕ ×ℕ is countable, it is enough to show that there is an injection from ℕ ×ℕ to ℕ.

Q: Prove the following by using Principle of Mathematical Induction for all n ∈ ℕ:

A: We will prove the given statement

Q: Prove by induction that for all n ∈ N with n ≥ 8, 2^n ≥ 3n^2 + 5

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Q: Use mathematical induction to prove the following ormulae for every positive interger n   1^3 +…

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