New York University Combinatorial and Algebraic Proofs Questions

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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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Basic Strategy

Combinatorial Proofs

algebraic proofs

binomial identity

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