Please help me with an explanation. Can you guide through this. I'm doing this for fun. All the information is here

662267eeb3470ecd... IQ I}; Math I1 P lem 2 January 31, 2020 1. Recall that if x is a set then X'\" denotes the set of all k-subsets of it. [If H c X'\" we sometimes call H a k-uniform hypergroph; each edge has It uertices.) We say that H 9 Km is bipartite if there is a 2-coloring of the elements ofx such that every k-set in H gets both colors. For each positive integer It greater than 1, let fik] be the smallest possible number of edges in a k-uniform hypergraph {on any vertex set X] which is not bipartite. {at Find f{2]. Explain briefly. {bl Show that f[3] s 10. {c1* Show that f[3] g T. [This part gets a * because it's not so easy.) to] Show that k] 2 2"" for each It 2 2. [Hint: Probabilistic method] 2. Find the fraction of entries in fan infinitel Pascal's triangle which are odd. Remarks: {at You might as well work mod 2. {bl On p. 2? of Cameron there are some useful comments aboUt self-similarity. {cl Since there are an infinite number of entries, you will have to define what \"fraction\" means. Give a precise definition. You might think in terms of the rows. {d} Using your definition, solve the problem. 3. Do #3 in Cameron p. 44. Do {at by two methods: [i] factorial proof (ii: combinatorial proof You might do [cl by induction. Induction on either n or kworks. but one is much easier. Skip td] since we did it in class. Typo on tel: it should say "if n is odd\". [Hint on tel: one way is to find a polynomial that can be written 2 different ways. then compare coefficients of the same power of x] 4. How many sequences of 12 letters of the English alphabet are there {at if repetitions are not allowed and the letters must be in alphabetical order to] if repetitions are allowed and the letters must be in alphabetical order {c.l if there must be precisely r1 vowels [repetitions allowed} and 8 consonants (repetitions not allowedl. [The English alphabet has 5 vowels and 21 consonants] 5. Simplify {so find the sum]: in It. 1 4n k. {3-1 Ei'lahb [ll-12:20 {k)3 Do [h.: In two ways: [I] Write [1") in a different formI then add. (Ii: Integrate the binomial theorem, not forgetting the constant of integration. 6. Calculate the number of 12-permutations of {4A, 53, 3c} in two ways: {at "Direct method\" so your answer will be the sum of three multlnomial coefficients tb] Consider the number of 13permutations of this Multiset and find a useful bijection. to] Your answers must be equal. This might suggest an identity which ge nerallzes some other identity you have seen before? State the new identity