Discrete math

1. Consider the set ð‘†={(ð‘¥1,ð‘¥2,…,ð‘¥ð‘›)∈â„¤ð‘›|ð‘¥ð‘–≥0 ð‘Žð‘›ð‘‘ ð‘¥1+ð‘¥2+â‹¯+ð‘¥ð‘›=ð‘˜ð‘š}. Throughout the problem assume that ð‘˜,ð‘š∈â„¤ and ð‘˜≥2,ð‘š>0.

a. Assume ð‘˜=2. Describe a method of counting the number of elements in ð‘† whose entries are all even. (that is, explain how to count the number of ordered non-negative solutions to ð‘¥1+ð‘¥2+â‹¯+ð‘¥ð‘›=2ð‘š in which all the ð‘¥ð‘– are even)

b. Assume that ð‘š≥ð‘›≥3. If you randomly select an element of S, what is the probability that its coordinates are all equivalent ð‘šð‘œð‘‘ ð‘˜?

Hint: there are two distinct cases to consider – ð‘˜|ð‘› and ð‘˜âˆ¤ð‘›.

2. Consider the set of vertices, ð‘‰={1,…,ð‘›}. The set ðºð‘›={(ð‘–,ð‘—)∈ð‘‰×ð‘‰|ð‘–<ð‘—} represents a graph. (recall the definition from class that an ordered pair represents an edge between ð‘– and ð‘—)

a. For which values of ð‘› does ðºð‘› contain an Eulerian circuit?

b. How many distinct Hamiltonian circuits does ðºð‘› contain?

Hint: it might be more intuitive to count Hamiltonian paths and then investigate how many circuits each of these paths can extend to.

3. Let ð‘ƒ={ð‘∈â„• | ð‘>2 ð‘Žð‘›ð‘‘ ð‘–ð‘“ ð‘=ð‘Žð‘ ð‘¡â„Žð‘’ð‘› ð‘Ž=1 ð‘œð‘Ÿ ð‘=1}.

a. Find the equivalence classes of ð‘ƒ under congruence ð‘šð‘œð‘‘ 4.

b. If ð‘∈ð‘ƒ and ∃ð‘¥,ð‘¦∈â„¤ such that ð‘=ð‘¥2+ð‘¦2, which of the equivalence classes listed in part a) is ð‘ an element of?

Hint: recall that if ð‘Ž≡ð‘ ð‘šð‘œð‘‘ ð‘› and ð‘≡ð‘‘ ð‘šð‘œð‘‘ ð‘› then ð‘Ž+ð‘≡ð‘+ð‘‘ ð‘šð‘œð‘‘ ð‘› and also ð‘Žð‘≡ð‘ð‘‘ ð‘šð‘œð‘‘ ð‘›.

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