Quick Homework Help

0 people bookmarked this map

# Precalculus MM: A New Day, The Ancient Queen: Topics in Discrete Math 2: NWN 0153c

Note on indistiguishability: If the elements of the set of all elements under consideration belong to two or more groups (e.g. boys and girls), then permutation can take place for the elements comprising each group if each element within a group is able to be distinguished from all the other elements of the same group (e.g. no two boys look exactly the same and no two girls look exactly the same). If the elements comprising a particular group are indistinguishable (e.g. identically composed red and blue balls of the same size), then no permutation can take place for the elements comprising each group as there will be no way to tell one element from one group from another element from the same group, i.e. 0!x0! = 1 for these red and blue balls. The only permutation that can take place is among the groups themselves, not the elements comprising the groups, i.e. red ball first then blue, then red, then blue and so on, or blue ball first then red, then blue, then red etc. The situation for these two distinguishable groups of indistinguishable elements per group is therefore 0!x0!x2 = 2.
++++++
Eltoora has chosen 12 sections out of 208 from The Chronicles of Shar she considers important Zanzi read. Barring a few minor cross-references relating to certain individuals and the chronology of certain events, each section is self contained and it is therefore not strictly necessary for Zanzi to concern herself too much about the order in which these 12 sections should be read. ("The Ancient Queen" is actually the penultimate section listed in Eltoora's note). A question on combinations: If Zanzi decides to read 3 sections during each reading session, how many different combinations of these groups of 3 sections are there from the set of 12 sections? Because each section is independent and the order in which they are read within the group of 3 sections doesn't matter, the permutation of these 3 chosen sections, designated r, needs to be taken into account when calculating the solution. 12P3 will permute each of the 3 sections so that the first three factors of 12! will occupy the three r placeholder values, as in nPr. However 12.11.10 includes the permutation of the three sections chosen by Zanzi to read. But the order Zanzi reads these three sections doesn't matter, so to compensate for this divide nPr by r!, which has the effect of removing from the total permutation the section permutations to be disregarded. So there are (12P3)/3! different combinations of the groups of 3 sections from the set of 12 = 220. Or, to put it another way, there are 220 different ways a subset of 3 elements can be permuted from a set of 12 elements.
++++++
Notes on probability of multiple events: If the probability of events occurring are mutually exclusive, such as rolling a die to get a 5 or even number, then P(A or B) = P(A)+P(B). If the events are not mutually exclusive, such as rolling a die to get a 5 or odd number, then P(A or B) = P(A)+P(B)-P(A and B). Of the 12 sections from The Chronicles of Shar Eltoora has chosen for Zanzi to read, 7 relate only to Shar and 2 relate to both Shar and Celeste, one relates only to Celeste and 2 relate to deities in general. If Zanzi chooses a section to read at random from Eltoora's list, then the probably that this section will in some way relate to Shar, noting that Shar is of course a deity, is given by P(S)+P(S&C)+P(D+S+S&C)-P(S)-P(S&C) = 7/12+2/12+(2+7+2)/12-7/12-2/12 = 11/12. The probability that the section will relate to either Celeste or deities is given by P(C)+P(S&C)+P(S)+P(S+D+S&C)-P(S)-P(S&C) = 1/12+2/12+7/12+11/12-7/12-2/12 = 12/12 = 1. The probability that the section will relate to either Shar or deities = P(S)+P(S&C)+P(D+S+S&C)-P(S)-P(S&C) = 7/12+2/12+(2+7+2)/12-7/12-2/12=11/12. The probability that the section will in some way relate to Celeste is given by P(S&C+C)+P(C)-P(C) = (2+1)/12+1/12-1/12=3/12=1/4.