Consider the following technique for shuffling a deck of n cards: for any initial ordering of the cards, go through the deck one card at a time and at each card, flip a fair coin. if the coin comes up heads, then leave the card where it is; if the coin comes up tails, then move that card to the end of the deck. after the coin has been flipped n times, say that one round has been completed. for instance, if n = 4 and the initial ordering is 1, 2, 3, 4, then if the successive flips result in the outcome h, t, t, h, then the ordering at the end of the round is 1,4,2,3. assuming that all possible outcomes of the sequence of n coin flips are equally likely, what is the probability that the ordering after one round is the same as the initial ordering?

Acaterer knows he will need 60, 50, 80, 40 and 50 dinner napkins on five successive evenings. he can purchase new napkins initially at 25 cents each, after which he can have dirty napkins laundered by a fast one-day laundry service (i. e., dirty napkins given at the end of the day will be ready for use the following day) at 15 cents each, or by a slow two-day service at 8 cents each or both. the caterer wants to know how many napkins he should purchase initially and how many dirty napkins should be laundered by fast and slow service on each of the days in order to minimize his total costs. formulate the caterer’s problem as a linear program as follows (you must state any assumptions you make): a. define all variables clearly. how many are there? b. write out the constraints that must be satisfied, briefly explaining each. (do not simplify.) write out the objective function to be minimized. (do not simplify.)

Astrid spent $36 on new shoes. this was $12 less than twice what she spent on a new skirt. part a: which equation would solve for how much she spent on the skirt? part b: solve the equation in the workspace provided below. how much did astrid spend on her skirt?

The sum of two numbers is 0. twice the smaller number subtracted from 3 times the larger number is 10. let x represent the larger number and y represent the smaller number. what is the equation