These problems involve a jelly bean factory that makes 5 different colorsof jelly beans: blue, green, orange, purple and red. They randomly combine the jellybeans of different colors into bags of 10 jelly beans eachl. Required:
a. How many different color combinations of jelly beans are possible for the bags?
b. How many different bags of 10 jelly beans are possible if there are at most 4 purple available?(The factory is almost out of purple jelly beans.)
c. How many different bags of 10 jelly beans are possible if there are at most 4 purple and at most 4 orange?
d. You need to get jelly beans for your 7 friends, one for each. But to keep the peace, you need to give them all the same color jelly bean. Since the colors of the jellybeans are random (you don’t get to pick) how many individual jelly beans must you have be certain that you have at least 7 that are the same color?

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.