.... Consider all permutations of the set {1, 2, n}. Prove that the average number of fixed elements in a permutation of the set -1 (n-1)dn-k (where dn is the number of 1 {1,2,...,n} is equal to k=1 (n-1)! of n elements). derangements

Q: For any two non-negative integers x and y, let d(x, y) = x + y. Let x, y, z be any three…

A: See solution below

Q: Prove that (n – 1) +n° + (n + 1)° is divisible by 9 for n E {1,2,3, . }

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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: Find a closed formula for S(n, n − 2), for all n ≥ 2.

A: Find a closed formula for S(n, n − 2), for all n ≥ 2.

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: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: 15. Let z = 2cos0 + i(1 – 2sin0) where -n < 0 < n Show that Iz – il = 2

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Q: Let S= {n | n=5 k+ 4 for some integer k} Let T= {n|n=5 j– 1 for some integer j}

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Q: (b) For any n ≥ 1, the factorial function, denoted by n!, is the product of all the positive…

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Q: n Exercise 2.11. Prove that n = [2] + 10 for all n E Z.

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Q: Use sigma notation to write the sum. [¹(-²) + ³] + [¹( ² ) + ³] + . . . + [4 (²) + ³] i = 1 X

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Q: (a) Let n, k € Z with 0 ≤ k ≤n. Write down the formula for the binomial coefficient (7) and use it…

A: a) Given: n,k∈ℤ with 0≤k≤n To prove that  nk+nk+1=n+1k+1

Q: Let the universe U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 3, 4}, B = { n |n is odd}. Find %3D (AUB)'…

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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: 4. Show that P(n,k)= P(n – 1, k)+ k · P(n – 1,k – 1). Using Combinatorial Proof.

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Q: 5. Use the binomial theorem to prove that C(n, 0) C(n, 1) + C(n, 2) + (-1)"C(n,n) = 0. —

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