For each positive integer n, consider the pair (3^n −1, 5^n −1). For n = 1, this is the pair (2, 4) and for n = 2, it is (8, 24). Can you find a value for n such that both numbers in the pair (3^n − 1, 5^n − 1) are divisible by d = 7? Can you find more than one such n? Do you think there are finitely or infinitely many n such that 3n − 1 and 5n − 1 are both divisible by d = 7? Why? Explain your reasoning as carefully as you can. Now, what if we replace d = 7 with d = 11? Or with d = 13? What if we take d = 77

or d = 1001? Describe any patterns you think are interesting.

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.)