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I am expecting clear and complete explanations for each question. As for question 4, both combinatorial and algebraic proofs are required. Thank you.

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1. Let k E N. Let A be the set of all 2k-tuples (a1, …, Q2k) such that

• ai = – O for k values of i,

– 1 for k values of i, and

• for each i,

• ai

i

Σα; 2

j=1

(a) Find a bijection between A and some set B of shortest paths that

you think you can count. Justify that your function is a bijection.

(b) Find |Bl.

2. An elementary school has n classes with 20 students each. In how many

ways can the school choose a team of k students in such a way that at

least one class has multiple team members?

3. You have 200 different-looking tiles. (Each is a different solid color. You

only have one tile of each color.) You sell trays that are made by lining

up 5 tiles in a row and gluing them to a backing.

A customer orders 4 trays. In how many different ways can you fulfill the

order?

4. For each of the following, give two proofs: one algebraic and one using

counting. .

(1 + 1)2 = 31-1(2n + 3)

(α) Σ(1)

Σ

r=0

(b)

( (0, 0, 0, n.)a

23%.

A1, A2, A3, A4,

α1 ,+a2+13+α4=n

(Here n is fixed, and the sum is over all 4-tuples a1, 22, 23, 24 of

nonnegative integers whose sum is n.)

5. Let n be a positive integer. Define a permutation partition of [n] to be a

set of nonempty permutations of elements of [n] such that each element

of [n] appears in exactly one of the permutations. Each of the permuta-

tions is called a part of the permutation partition. For instance, here is a

permutation partition of [7] with 3 parts:

{1436, 2, 75}

and here is a permutation partition of [7] with one part:

{7426513}

Let t(n,r) denote the number of permutation partitions of [n] with exactly

r parts.

(a) Use a counting argument to get a recurrence relation for t(n,r) that

expresses t(n,r) for positive r and n in terms of r, n, tín – 1,r),

t(n,r – 1), and t(n – 1, r – 1). (You don’t have to use all of these,

but you can’t use just r and n..) Your answer should have at most 4

summands.

(b) Use a different counting argument to get a formula that expresses

t(n,r) for all r and n in terms of r and n only.

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Tags:

Basic Strategy

Combinatorial Proofs

algebraic proofs

binomial identity

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