Consider the simple pendulum x_ 1 = x2; x_ 2 = b ml2 x2 g l sin (x2) : Using the total energy as the candidate Lyapunov function, show that [00] is an asymptotically stable equilibrium point. Find an estimate of the region of attraction of [00] so that the projection of the region of attraction on the x1 dimension includes the interval (pi; pi)

Afriend of mine is giving a dinner party. his current wine supply includes 10 bottles of zinfandel, 8 of merlot, and 11 of cabernet (he only drinks red wine), all from different wineries. (a) if he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this? ways (b) if 6 bottles of wine are to be randomly selected from the 29 for serving, how many ways are there to do this? ways (c) if 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety? ways (d) if 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? (round your answer to three decimal places.) (e) if 6 bottles are randomly selected, what is the probability that all of them are the same variety? (round your answer to three decimal places.)