Let dg denote the number of derangements of order k. (a) Prove the identity C) d = n! with a combinatorial argument. (b) Substitute k!/e for d in this identity and compare the two sides for large n.

Q: 2. n³+3n²-4 = 0 (n³)

A: We have used definition

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Q: Evaluate L{r'e}. (a-b)! A. aーb+1 (a+b)! В. a+b+1 b! C. (s -n+a) b+1 a! D. (s-b+r)** (ab)! E. ab+1

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Q: 4. Use induction to prove that n 1 Σ(i)(i!) = (n+1)! - i=1

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A: The combination formula is defined as (nr)=n!r!(n−r)! Therefore…

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Q: 2. Using induction, prove that 1 %3D i(i+1) n+1

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Q: f(n)={1 if n is odd; n if n is even  Prove or disprove. f(n) is O(n).

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Q: + メ dx 1+ I Ans: (am)! ] %3B リx」 + C, イ

A: Consider the differential equation, y"+xy'+y=0   ⋯⋯⋯1. To show that the solution for the above…

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Q: 24). Legendre Polynomial P2 (x) · ......... (A) x? (B) 1/2 (x² + 1) (C) 1/2 (3x² - 1) (D) 1/2 (3x² +…

A: We know the legrendre  polynomial expansion.

Q: (b) Prove that for every n e Z, n³ +n is even. (c) Prove that for all ne Z, n is odd if and only if…

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Q: 3. Let f(n) be defined for n >0 be defined recursively by f(0) = 7, f(1) = -5, f(n + 2) = (f(n) –…

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Q: A container contains 35 green tokens, 5 blue tokens, and 3 red tokens. Two tokens are randomly…

A: There are total (35+5)=40 non-red tokens and 3 red tokens in the container.  And total (35+5+3)=43…

Q: For any positive interger n, (n + 1)! / n! - n = (A) 0 (B) 1 (C) n (D) n + 1 (E) n!

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Q: Find {2), f3), R4), and {5) if fis defined recursively by f0) = -1, A1) = 2, and {(+1) = Jtn}*s(n –…

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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…

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A: We have, an = 212nWe have to find a recursive definition with initial conditions of an starting from…

Q: An = 2n an = 3n – 5 an = (n + 1)/3 an = v2

A: Given:  a) an=2n b) an=3n-5 c) an=n+13 d) an=2

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A: The problem can be solved using concepts of permutation and combination.

Q: Prove that n Sk k=1 = n(n + 1)(2n + 1) 6 for all n € N.

A: To prove:We will use mathematical induction to prove this.Let

Q: 2, +x2 dy Ant ye G(は ) %3B +5(メ+ を ナ

A: Given that x3-1y''+x2y'+xy=0 Take  y=∑n=0∞cnxn=c0+c1x+c2x2+c3x3+c4x4+c5x5+c6x6+c7c7+. So the…

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A: T(n) is the divisor function which counts the number of divisors of a number.

Q: Prove gcd(n, n+1) = 1.

A: To prove the below expression. gcdn, n+1=1

Q: (1) > 1 <2- i=1

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Show that for n ≥ 3, ô(n) is even.

A: We have from the definition of Euler ϕ function: ϕ(1)=1 and ϕ(2)=1 Now Let n ≥ 3

Q: (d) If each fn has fewer than M discontinuities (where M ∈ N is fixed), thenf has fewer than M…

A: To Show: If each fn has fewer than M discontinuities (where M ∈ N is fixed), then f has fewer than M…

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A: The given series .The aim is to compute the sum of the series.

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

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Q: Prove Γ (n) = (n-1)! where n ∈ ℕ

A: Concept: The proof for this gamma integral will be done using mathematical induction