2. Let Qn be the number of permutations of {1,2, .., n} in which none of the patterns 12, 23, .., (n – 1)n оссurs. ) Find Q3 by listing all the possible permutations. ) Use the inclusion-exclusion principle to prove that (a) { (b) (- - n - 2 + 2! (-1)"-1 (п — 1)! n – 1 п - 3 Qn = (n – 1)! (n- 1! 3!

Q: 5. Verify if the following are difference sets and state v, b, k, r, A for the ones that are: a)…

A: 5(a) Given that, (0,8,15,17,20,21)mod31 The cardinality of the set is 6 with mod31. Here V is the…

Q: 5. Let 02(n) be the sum of the squares of the divisors of n 0₂(n) = Σd². din Make a guess at a…

A: It is given that ,    then

Q: Show that C(p,1), C(p,2),..., C(p,p-1) are all divisible by p if and only if p is a prime.

A: As per the question we have to prove that C(p,1), C(p,2),..., C(p,p-1) are all divisible by p if and…

Q: Let n ≥ m be positive integers. Prove that Σ(*) (*) = (m)2n-m. (Hint: This can be solved with a…

A:

Q: 1. A restaurant is preparing a large round table with 2 +1 seats (the seats are unla- belled). The…

A: Given, Number of available seats = 2n+1 Number of red napkins = n+1  Number of blue napkins = n

Q: Given a positive integer n, let a2,n–1 a2,n a3,n-2 a3,n-1 a3,n An = аn-1,2 аn-1,n-2 аn-1,п-1 @n-1,п…

A: Given: Here the given positive integer n, Let An = 00...00a1,n00...0a2,…

Q: 7 Disprove the following statements: (a) If n and k are positive integers, then n-n is always…

A: Given Data: To disprove the following statements.

Q: 5-(1,5.25, 125.62s,3125)

A: Given,

Q: If a set S has n > 3 elements, how many subsets of S have more than one element, but less than n…

A:

Q: Write each as a single factorial. i) 90*8! ii) (n² + n)(n 1)! iii) (n +1)!/(n -1)!

A:

Q: 6. Use the binomial theorem to prove that 3 = C(n,0) + 2C(n,1)+2²C(n, 2) + +2"C(n, n)

A: By using the binomial thorem to prove that, 3n=Cn, 0+2Cn, 1+22Cn, 2+...+2nCn, n

Q: Prove that n! (n-r)!r! (4) is the number of subsets (combinations) with r elements which can be…

A:

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

A:

Q: one of the two. How do we find the number of individuals who enjoy both coffee and tea?

A: 6. In a group of 60 people. 27 prefer cold drinks and 42 prefer hot drinks, with each participant…

Q: Use inclusion-exclusion to find the number of ways to arrange the six numbers 1, 2, 3, 4, 5, 6 such…

A: please see the next step for solution

Q: Let a = (1 73)(5 49 2) and B = (2 3)(7 4)(5 18) be elements of S10. Then, a B is O (12974)(3 58) O…

A: Let α=1735492 and β=2374518 be elements of S10, then α2β-1 is

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Q: Let o = (143)(2576) and T = (17)(253)(46) be permutations in S7. Compute (a) то (b) OT (c) o²

A:

Q: y = C¡e e-2= + Cze³* + C2e%

A:

Q: b)+++ -1- for all n 1. ... 4 2"

A:

Q: Prove that, for all integers  , K>=5…

A: Note: There is minus sign instead of plus. And the proof us shown below.

Q: a)Show by one combinatorial method that: 2 2 n C) - () - (") 2n +... + 2 n

A: Given information: The following identity is provided: n02+n12+n22+................+nn2=2nn It is…

Q: Remark: For positive integers n, m, we have • (n) n (m) (n) + (m) = = (lcm (n, m)) (gcd (n, m))

A:

Q: 1) What are the values of C and k of the term Cykx20 in the polynomial expansion of (x² + y )100? 2)…

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

Q: Which of the following statements are true?   A. There are \binom{3}{2} ways to select two letters…

A: The objective is to identify the true statements.

Q: One solution of the recursion relation an+2 = n+I» n 2 0, is n+1 » a) a2k+1 = (-1)*a 2k k! k 2 0 b)…

A:

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A: English alphabets vowels are A,E,I,O,U and rest 21 are consonants

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Q: 50° S. 80° R 60° 100 7. 7.

A: The measure of the arc length is double than the angle formed by that arc to the circumference of…