Question

Q1: An environmental group order some detection chips from a nanofabrication company. nanofabrication company links these 9 probes step by step and organized in a straight line. when probe C and E are nearby. Other probes work well without restrictions. They want 9 different probes (A B CDEFGH I) on the chip for their detection. The However, probe A functions only when probe B is nearby, and probe D functions only 1. Suppose the nanofabrication company links 9 different probes in random order along a straight line. What is the probability that the chip is functional? 2. What if they link the 9 probes in a 3 x 3 square dot matrix?

Answer 1

1.

a prabes can be atran sged dn a straiaht line n Wos D funebiens onlj when and E ore nesrby รื่ D functions only when the arrangement is CDE or EDC A functions nbj when B neavby → A Run ctions onl when the arrangement ò AB or BA e drvanaement S Thuus the chip is funchionol onby when AB or BA and CDE or EDC bolh are presentnthe straght line Consider the case where AB and CDE are present Tn the stroight line Assuming AB and CDE to be. bw single probes each 5 there e six distinct probes whte con be arrenaed in a strai Une n 6! woS There ore four smilar conditcons 5 each o AB and cDE AB and EDC, BA ord CDE, BA and £DC Thus the chip ts Functional 4x 6 combinat iens of the probes 7 - ProbDL cllon Possible 4x6

2.

GSe l Top row s CDE For one combination ot A ond l possible combinabions: AİB AİB AİB AİB Au these cembdnabtons ore posscble for another cembunatcon of A ond B (Interchanging positions of A and B) A these combnoions are possibla when CDE ts top All these. cembinotions ove possible or EDC

Case II For one cembinotien of A and oll possible AID DÍA CBE CA E Au these combdvnabdons Qre posscble fo onother cembinatcon of A and B (Interchanging positions of A and B) All these combinations are possible when CDE ts rernaed as All these cembinations ove possible er EDc :Total no p ossib le cembonat ions-7x2xx2-11 OSS

Cose IX CD ïsorron ged as For one cembinatten of A and B > oll possible AID DIA BID DİB AIB BID DIB Au these cembövvabtons Ore possible for another cemünahvon of A- and B (Interchanging positions of A and B) All these combinatiens are possible uhen CDE is All these combinottans are possible Per EDC