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.
The solution gives detailed steps on solving a couple of discrete questions. The topics include Binomial Theorem, coefficient of terms, reflexive property, antisymmetric property, transitive property and partial order.
Please see attached file with questions and provide ALL work so that I can understand how you arrived at the answers. Thank you.

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 th
Looking for some answers to discrete math questions. Must show work so that I can understand how you achieved the results.
See attached.
1. Note the contrapositive of the definition of onetoone function given on of the text is: If a ≠ b then f(a) ≠ f(b). As we know, the contrapositive is equivalent to (another way o
1. After a weekend at the Mohegan Sun Casino, Gary finds that he has won $1020—in $20 and $50 chips. If he has more $50 chips than $20 chips, how many chips of each denomination could he possibly have?
2. If there are 2187 functions f : A→B and B = 3, what is A?
3. Let A ⊆ {1, 2, 3, . . . , 25} where A = 9.
Logic & Set Theory; Boolean Algebra; Relations & Functions
1. How do we distinguish relations from functions?
2. What sort of relation is friendship, using the human or sociological meaning of the word? Is it necessarily reflexive, symmetric, antisymmetric, or transitive? Explain why it is or is not any of these. What othe
Please help with the following proofs. Answer true or false for each along with step by step proofs.
1) Prove that all integers a,b,p, with p>0 and q>0 that
((a+b) mod p)mod q = (a mod p) mod q + (b mod p) mod q
Or give a counterexample
2) prove for all integers a,b,p,q with p>0 and q>0 that
((ab)mod p) mod q=0
Question 1:
When is the mean the best measure of central tendency? When is the median the best measure of central tendency? Explain.
Questions 2:
A researcher has determined that the distribution of annual salaries of NBA players is bellshaped and symmetrical about the mean salary, do you think that introducing Michael