combinatorics Problem 7: You have n stones. You break the stones into some number of groups,...
combinatorics
Problem 6: A baker has n cupcakes in a line. They break the line into some number of sublines, then they put two (identical) cherries on two cupcakes from each subline, and finally a quality control robot marks each subline as good or bad. Let fn be the number of ways this can happen. Find the ordinary generating function of f
Problem 6: A baker has n cupcakes in a line. They break the line into some number of...
Assume you have a necklace of stones. Some of the stones have positive value and some have negative value. You have the opportunity to snip the necklace in two places (creating two bands) and weld the endpoints of one of the two bands back into a necklace. You would like your new necklace to be as valuable as possible. You can assume the necklace has n stones with values O,,vn-1. (1) Give an efficient algorithm to find the value of...
Question 7: Consider 10 male students M1, M2,... , Mo and 7 female students F1, F2,... , F Assume these 17 students are arranged on a horizontal line such that no two female students are standing next to each other. We are interested in the number of such arrangements, where the order of the students matters Explain what is wrong with the following argument We are going to use the Product Rule: » Task 1: Arrange the 7 females on...
PartB (COMBINATORICS) -LEAVE ALL ANSWERA IN TERMS OF C(n,r) or factorials, Q4(a)(i ) In how many ways can you arrange the letters in the word INQUISITIVE? in how many of the above arrangements, U immediately follows Q? Q4. (b)Su next semester. Your favorite professor, John Smith, is teaching 2 courses next semester and therefore ppose you are a math major who is behind in requirements and you must take 4 math courses you "must" take at least one of them....
5. Identify the following: k (number of groups or samples) N, (number of cases within each group or sample) N (total number of cases) df (between-groups degrees of freedom)= How is it calculated? df (within-groups degrees of freedom) How is it calculated? critical value for F (using the F table) SSB (Sum of Squares Between groups) = ssw (Sum of Squares Within groups) Mean square between= How is it calculated? Mean square within= How is it calculated? F ratio (test...
Part B(COMBINATORICS) LEAVE ALL ANSWERA IN TERMS OF C(nr) or factorials Q4(a)6) In how many ways can you arrange the letters in the word INQUISITIVE? in how many of the above arrangements, U immediately follows Q? Q4. (b)Suppose you are a math major who is behind in requirements and you must take 4 math courses and therefore next semester. Your favorite professor, John Smith, is teaching 2 courses next semester you "must" take at least one of them. If there...
06. Do any two of the following three parts Q6(a). Solve the following recurrence relation; Q6(b). Find a recurrence relation for an, which is the number of n-digit binary sequences with no pair of consecutive 1s. Explain your work. Q6(c) Solve the following problem using the Inclusion-Exclusion formula. How many ways are there to roll 8 distinct dice so that all the six faces appear? Hint: Use N(A'n n. NU)-S-,-1)' )-S-S2+S-(-1)Sn U- All possible rolls of 8 dice, Aj-Roll of...
This problem is concerned with evaluating some improper integrals. In particular you will use an improper integral over an interval of infinite length to evaluate an integral of a function not defined at one end point. This will involve a special function「which arises in many applications in the sciences. a. Evaluate Jo (log z) dr. b. Explain how you would evaluate Jr*(log x)7 dr, but do not actually compute it. Would your method work if the exponent'8, were replaced by...
Q3 (Due Wednesday 11 September—Week 7) Let (G, *) and (N,) be groups. Suppose that g Ha, is a homomorphism from from G to Aut(N)—that is, suppose that a, o ah = agh for all g, h E G. Let N a G denote the set N X G, and define a binary operation • on N a G by (m, g) + (a, b) = (m + ag(m), g * h). (1) Prove that (N a G, is a...
Problem 2: Let pı .. .pn E Sn be a permutation, considered in its one-line notation. A descent in p is an index 1 < i S n 1 such that p, > Pi+1. How many permutations in Sn have exactly one descent?
Problem 2: Let pı .. .pn E Sn be a permutation, considered in its one-line notation. A descent in p is an index 1 Pi+1. How many permutations in Sn have exactly one descent?