There are five students $$S1$$, $$S2$$, $$S3$$, $$S4$$ and $$S5$$ in a music class and for them there are five seats $$R1$$, $$R2$$, $$R3$$, $$R4$$ and $$R5$$ arranged in a row, where initially the seat Ri is allotted to the student Si, i $$=$$ $$1$$, $$2$$, $$3$$, $$4$$, $$5$$. But, on the examination day, the five students are randomly allotted the five seats.The probability that, on the examination day, the student $$S1$$ gets the previously allotted seat $$R1$$, and NONE of the remaining students gets the seat previously allotted to $$him/her$$, isFor i $$=$$ $$1$$, $$2$$, $$3$$, $$4$$, let Ti denote the event that the students Si and $$Si+1$$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $$T1$$∩$$T2$$∩$$T3$$∩$$T4$$ is

Question

There are five students $$S1$$, $$S2$$, $$S3$$, $$S4$$ and $$S5$$ in a music class and for them there are five seats $$R1$$, $$R2$$, $$R3$$, $$R4$$ and $$R5$$ arranged in a row, where initially the seat Ri is allotted to the student Si, i $$=$$ $$1$$, $$2$$, $$3$$, $$4$$, $$5$$. But, on the examination day, the five students are randomly allotted the five seats.The probability that, on the examination day, the student $$S1$$ gets the previously allotted seat $$R1$$, and NONE of the remaining students gets the seat previously allotted to $$him/her$$, isFor i $$=$$ $$1$$, $$2$$, $$3$$, $$4$$, let Ti denote the event that the students Si and $$Si+1$$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $$T1$$∩$$T2$$∩$$T3$$∩$$T4$$ is

Infinity Learn · Accepted Answer

Total number of cases $$=$$ $$5$$!Since $$S1$$ gets seat $$R1$$ and none of the other gets previously allotted seat, we have derangement of $$4$$ students.So, required $$probability=4$$!$$1$$−$$11$$!$$+12$$!−$$13$$!$$+14$$!$$5$$!$$=9120=340Event$$ $$T1$$∩$$T2$$∩$$T3$$∩$$T4$$ : No two students with consecutive index sit adjacent.So, we have following possible arrangements:$$13524$$,$$1425324135$$,$$24153$$,$$2531431425$$,$$31524$$,$$35142$$,$$3524141352$$,$$42531$$,$$4251352413$$,$$53142Thus$$, favorable cases $$=$$ $$14Hence$$, required $$probability=14120=760$$

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