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Steps on Solving Discrete Questions

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I need some help with this mathematical questions:
(a) Use the Binomial Theorem to write the expansion of (x + y) 6?
(b) Write the coefficient of the term x2y4z5 expansion of (x + y + z) 11.

Let A = {a, b, c, d}, and let R be the relation defined on A by the following matrix: (see attachment for
(a) Describe R by listing the ordered pairs in R and draw the digraph of this relation
(b) Which of the properties: reflexive, antisymmetric and transitive are true for the given relation? Begin your discussion by defining each term in general first and then how the definition relates to this specific example.
(c) Is this relation a partial order? Explain. If this relation is a partial order, draw its Hasse diagram.

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Solution Summary

The solution gives detailed steps on solving a couple of discrete questions. The topics include Binomial Theorem, coefficient of terms, reflexive property, anti-symmetric property, transitive property and partial order.

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Steps on solving 4 discrete math questions

Please see attached file with questions and provide ALL work so that I can understand how you arrived at the answers. Thank you.

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1. Let P (n) be the statement that 13+ 23+···+ n3 =(n(n+ 1)/2)2 for the positive integer n.
a) What is the statement P (1)?

b) Show that P (1) is true, completing the basis step of the proof.

c) What is the inductive hypothesis?

d) What do you need to prove in the inductive step?

e) Complete the inductive step, identifying where you use the inductive hypothesis.

e) Explain why these steps show that this formula is true whenever n is a positive integer.

2. Show ALL work. Solve the following systems of equations using Gaussian Elimination. Your procedure should be in matrix form.
x1 + x2 + x3= 1
2x1 - x2 + x3= 2
-1x2 + x3= 1

3. Determine the solutions of the system of equations whose matrix is row equivalent to
Give three examples of the solutions. Verify that your solutions satisfy the original system of equations.

4. Solve for B: B=

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