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.

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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.

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

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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: 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)...…

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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: Eratosthene5-Legendre

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Q: Proof: For any n ∈ Z, 3 | (n3 - n)

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Q: 9. 2n (2) for for n ≥ 0

A: Solution