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: The number of permutations of n elements taken rat a time is denoted by nPr and is computed by or…

A: The number of permutations of n elements taken r at a time is given by nPr where nPr is given by,

Q: (a) Show that if n € N, then n+1 In dx. I

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Q: 2. + -+- 1.3 3.5 5.7 + + 1 n (2n-1)(2n+1) 2n+1

A: As per guidelines, I can do first question only.

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,…

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Q: A permutation of the set [10] = {1,2, 3, ..., 10} is a way of listing the elements of (10] in order…

A: We have to count the number of permutations of  [ 10 ] with certain properties.

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: (10) A woman has six friends. Each evening, for five consecutive days, she invites three of them…

A: A women has six friends. Each evening for five consecutive days, she invited three of them for…

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: Question 6. We can use (C) to represent C(n, k). Show that (") is divisible by n if n is a prime…

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Q: Question 5. Given positive integers n and m, how many sequences (a₁, a2,...,an) of n integers are…

A: Given Information:The two positive integers .To find:The number of sequences  of  integers are there…

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

A: We are given the Vandermonde identity is n+ml=∑k=0lnkmk+m-l. We have to show that…

Q: Identify the formula for the number of permutations when r objects are selected from n objects.…

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Q: Select the correct choice that completes the sentence below. The formula for the number of…

A: We want tell you which one is the correct choice.

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: 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: n people are seated around a round table (n ≥ 3) and each person rolls a fair twelve-sided die. If a…

A: The probability of rolling a prime number on a twelve-sided die is 4/12 = 1/3. The probability of…

Q: Given a set of letters X = T, S, U (i) show all possible arrangements of the letters and (ii) count…

A: Given that: Given a set of letters X = T, S, U. To find: (i) show all possible arrangements of the…

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.…

A: Note :As per guidelines only first 3 subparts have been answered if you need answers for the rest of…

Q: Question 6. Given positive integers n and m, how many sequences (a₁, a2,..., an) of n integers are…

A: The number of sequences (a1 ,a2 , ..., an ) of n integers that  can be calculated using…

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…

Q: 9. 2n (2) for for n ≥ 0

A: Solution