Let n, r E Z with 1 < r < n. Give a combinatorial (that is, a counting argument) proof that · · C(n,r) = n · . C(n – 1,r – 1) by counting in two different ways the following quantity: the number of ways to select a subcommittee withr people from a committee of n people, where a chair of the subcommittee is chosen.

Q: Which of the following statement is true about the formula of permutation? A. 'n' is always greater…

A: Given problems related to permutation and combination.  We have to solve and select the correct…

Q: 2. Let S be the set [9] = {1.2..9}. 1 (D) How many derangements of S have 9 in the 2nd position and…

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Q: Continuous variables produce numerical responses that arise from a counting process.Answer 9…

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Q: is less than The following numbers are given in ternary form. Find the real number that each…

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Q: 2n+2 52n+1 3n+2, 22n+1 is divisible by 19 co dady M N [ day f04+ 204d (21)

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Q: Consider the numbers sn defined by sn = 1 + 3 + 5 + 7 + … + (2 * n + 1), for all n ∈ ℕ. That is, sn…

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Q: 4. Find the union and intersection of these families, and prove your answers. (a) A = {A, ne N},…

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Q: T F 54. If the integer n is divisible by both 7 and 9, it is divisible by 63.

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Q: 4b. Show that there exists i and j with i < į such that the consecutive sum p;+pi+1++p; is divisible…

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Q: QUESTION 7. Find the number of permutations of length 9 that satisfy the following conditions: (i)…

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Q: Find a natural umber N such that if |a – 5| < e <1 then |a² – 25| < Nɛ.

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Q: d. Prove that for all integer n ≥ 2. 3n > 2n + n²,

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Q: Using the binomial coefficients (2) = n n! (n-k)!k! (1) The sum first n integers: 1 + 2 + ... + n =…

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Q: Prove the following step by step with explanation please,

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Q: (a) How many different groups of 6 people could be selected from a group of 24 to go on a trip?…

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Q: (d) For which e E {0, 1, ..., 9), is 12e128 divisible by 8? Find all such e. 1

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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: (c) Find an element h of S10 such that hog (1 8 10 2 7 3 6 4), = and write it in two-line notation.…

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Q: T/F explain C. There are as many 231-avoiding permutations of length n with k descents as there are…

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Q: n- 5. Use the Vandermonde identity to show that (²₁)-(:)(:1). k=0 Hint: Using central symmetry,…

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Q: 5. Let a, b, n be integers with n >0 such that a % n = b%n. Which of the following statements is not…

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Q: Let ao = 0, a₁ = 6, and an Use the PCI to prove that an = 7 - 1 for all n E w. = 8an-17an-2 for all…

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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: 1. The factorial notation n! is defined for a positive number n as n! %3D п (п — 1)(п - 2)...…

A: Questions: 1. The factorial notation n! is defined for a positive number n as, n!=nn-1n-2...3210. 2.…

Q: B D Which of the following about Sn are correct? (i): Each element of Sn has an order n (ii): There…

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

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Q: Q7- A fair coin is continually flipped until heads appears for the 4th time. Let X denote the number…

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Q: 1(1!) + 2(2!) + ..+ n(n!) = (n + 1)! – 1. %3D 3.

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Q: 6. Suppose that 2 = 1, 2, 3, ... and that P(k) = p(1 - p)k-¹, ken. Let A be the odd numbers in 22.…

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Q: n(n+1) for all n EN, • Given that 1+ 2 +3+..+n = the sum 70 + 71+72 + .+ 100 = 2

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Q: Prove that for every integer n where n is greater than or equal to 3, P(n+1, 3) - P(n, 3) = 3P(n,…

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Q: Proof: For any n ∈ Z, 3 | (n3 - n)

A: We see,       n3 - n = n(n2 - 1)                 = n(n - 1)(n + 1)  Now, there may be three cases.…

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…