How many permutation g on {1,2,3,4} which satisfies g(1)=2 are possible? (Write the number you calculate in the text field)

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: Show that for all n> 14, n cents can be represented by some combination of 3-cent coins and 8-cent…

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Q: 3. Show by mathematical induction that, for all n2 0, the equality +++.+ = 2 – +2 holds. ... 2Th

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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: 5.2.8 suppose that a store offers gift certificates and denominations of $25 and $40. Determine the…

A: The given denominations are $25 and $40.Here, the given denominations are multiples of 5.The amounts…

Q: The distinct characteristic roots corresponding to the recurrence relation an=7an-1 -10an-2 are:…

A: The given recurrence relation is an=7an-1-10an-2. Rewrite the recurrence relation is an=7an-1-10an-2…

Q: Use mathematical induction to show that E"-, ar ark-1 a(1-r") for all integers n 2 1-r 1. (Assume r…

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Q: п 12. Show that Ë O)=)2"-". Using Combinatorial Proof. k=0

A: To prove: ∑k=0nnkkm=nm2n-m

Q: Suppose a, n E Z and 0 < a<n. Prove that )-(".")-(C ) n- 1 in = a a а — 1 in two different ways:…

A: We have to prove, na = n-1a+n-1a-1 First let us prove this using formula. Using Formula…

Q: Let $n$ be a positive integer. Among C(2n,0),C(2n,1),..., C(2n,2n), C(2n,n) is the largest. True…

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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: 10. Show that È k(C) = n2"-1. Using Combinatorial Proof. k=1

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Q: Compute f2,f3, and f-1 for each of the following permutations. f= (2, 7,4,3,5)

A: Given: f= (2, 7,4,3,5) To Compute: f2,f3, and f-1

Q: 5. Prove by the principle of mathematical induction. show that 1+2+2 +...+ 2" = 2" –1 for all non…

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Q: Find the smallest positive integer n, пп such that 0 and 0 are 2 coterminal. And give an example

A: Coterminal angles are the angles in the standard position having a common terminal sides.

Q: Determine the identity element of the set N={1,2,3,..} of positive integers with addition.

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Q: Suppose that n is an integer such that n≥3. Using a combinatorial proof, show that:

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Q: 4. Show that if n is a positive integer, then Σ {a₁,...,ak}{1,2,...,n} (Here the sum is over all…

A: Given, If n is a positive integer, then we have to show that ∑a1,.....ak⊆1,2,....,n 1a1a2....ak=n

Q: If you have n random real number in a ordered list {a_1,a_2,…,a_n} , what is the probability of…

A: Given n random real number in a ordered list {a_1,a_2....a_n} {a_1,...a_i} is a monotonically…

Q: ROBLEMS 2.4 Find gcd(143, 227), gcd(306, 657), and gcd(272, 1479). ntoin integers r and y satisfying…

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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: Prove that gcd(k; n) = gcd(n - k,n).

A: Greatest common divisor (gcd) :            If a , b are two integers such that atleast one different…

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: 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: X2/ Suppose sr=20142} and c-{ {o8} Enumcrate the cLass of all 6-tields Containing and give cce)

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Q: 2. Let f and g be any permutations. Show that the order of h = fgf- is the same as the order of g.

A: Here f and g are tow permutations  and we are given that h=fgf-1. We have to show that both f and g…

Q: 2. Prove existence and uniqueness of a p-

A: Introduction: Two lines l and m are perpendicular if there exists a point A that lies on both l and…

Q: Assume that n > 2 and x E A is a fixed element. How many unordered selections without repeats of k…

A: N&gt;2, X is the element A is a fixed element  How many unordered selections withouts repeats of  k…

Q: The specialization of structural induction for ______ or  _______is called mathematical induction.

A: That's easy.

Q: If f is.q field a hom- saplar monic polphomial in tay can be factored as q, product of monie primes…

A: Any non-zero non-unit element as a product of primes up to order and up to units. Now  If a≠0 and…

Q: 4. If G = (Z8,) then G has an element of order 5. *True *False

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Q: 2. Show that if a]b then a'|b² for all integers a and b (Symbols V, 3, A, V, V,A,→,-)

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

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

Q: If gcd(m,p) = 1 and p is prime, show that gcd(m,pk ) = 1 for all k≥ 1.

A: To show, If gcd(m, p)=1 and p is prime, show that gcd(m, pk) = 1 for all k≥1.

Q: (3) prove That () 3

A: We have to use z = x + iy.

Q: 6. Use mathematical induction to show that, Vn > 7, n! > 3".

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Q: MK18. Show that in any set of 9 integers chosen from the set {2,3, 4, ... , 21, 22}, there are at…

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

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Q: 16. Prove that, for integers a and b, gcd(a, a + b) gcd(a, b). [Hint: Mimic the proof of Lemma…

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Q: 2) Let X be a finite set. Show by induction on n = = [X| that %3D |P(X)| = 2X1.

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Q: [Hint: It suffices to show that the numbe 11. Verify that 0, 1, 2, 2², 2³, . , 2° form a complete…

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Q: Show that the permutation [n, n-1, ... , 2, 1] has n(n-1) inversions.

A: An inversion in a permutation is a pair of numbers such that the larger number appears to the left…

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: (a) gcd(q, n) = 1 if and only if kq # 0 mod (n) for all 1 < k< n. %3D

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Q: Apply Mathematical Induction to prove that 3* > x² that for X>=1

A: first we will show that statement holds for initial value x=1 second we will assume that statement…

Q: Let a and b be positive integers such that ax + by %3D 7 for some integers x and y. Then O…

A: Greatest common divisior:  Let a and b be two integers such that gcd(a,b)=d if  d|a and d|b If…