Suppose a, n E Z and 0 < a

Q: The permutations p= p and q in Se have the two-line symbols 1 2 3 4 5 6 4 621 5 3 and q= 1 2345 1 5…

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Q: (9) Let n and k be positive integers. Recall from lecture that a weak composition of n into k parts…

A: 9.For  be positive integers, a weak composition of  into  parts is a -tuple  of nonnegative integers…

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: Births of boys with n = 20 births and p = 0.512 for a boy. In P(fewer than 8 boys)

A: Given, n=20 p=0.512 Normal as approximation to binomial will be given by as follows: X~Nnp,…

Q: The factorial of a positive integer n, denoted n!, is equal to the product of all integers between 1…

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Q: Prove that for all n € N, 91(4" + 6n + 8) and n € N, 31(n³-n).

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Q: rove that if r < s < n, then P(n, s) is divisible by P(n,r).

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Q: Prove that n < 2n for all n ∈N

A: We prove n&lt;2n by principle of mathematical induction Principle of mathematical inductionStep 1:…

Q: Prove that µ(d) p(n) = n . P ulp

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Q: 4g - 2x2 + 5% -1- Σακ(π)κ. 4x4 k>0 b) -(m)2 + (r)1 + 6 = Σb k>0

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Q: Let m%3= i=1 where pi 0 are integers then i=1 i=1 gcd(m, n) = I min(ai,bi). %3D i=1

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

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: Prove that for all n in J, 1 + 3 + 5 + • • • + (2n - 1) = n2

A: We have to prove 1+3+5+....+2n-1=n2

Q: et p(k) be the number of permutations of (1,2,3,4,5,6) such that there are actly k numbers remain as…

A: A fixed point in a permutation is an element that remains in its original position after the…

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…