Zadanie 9b, Paper 2, maj 2024, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2024, TZ1, Paper 2 (zadania wzorowane), pytanie 9(b). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
Ten distinct researchers occupy ten fixed, labelled seats around two separate circular workstations, five seats per circle. Adjacency means consecutive seats on the same circle, including the wraparound pair; seats on different circles are never adjacent. Assignments differing by a rotation are distinct because the seats are fixed. Researchers A and B must be adjacent.
Polecenie
Researchers C and D must also be nonadjacent, while A,B remain adjacent. Find the number of assignments.
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Odpowiedź z klucza
Subtract assignments whereboth pairs are adjacent. Once the edge forA,B is selected, the same circle has2 disjoint edges available for C,D and the other circle has5. Thus there are 10·(2+5) ordered choices of disjoint edges,2² internal orders, and 6! arrangements of the remaining researchers. Forbidden =10·7·4·6!=201600. Allowed =806400−201600=604800. Alternatively, after placingA,B, C,D have 8·7=56 ordered seat choices, of which 14 are adjacent, leaving 42; total 20·42·6!=604800.
Schemat punktowania
M1:subtract the both-adjacent case or count permissible placements directly. A1:correct same-circle/disjoint contribution or the 7 available disjoint edges. A1:correct opposite-circle/internal-order/remaining-seat combination giving 201600 forbidden or 42 allowed ordered choices afterA,B. A1:604800. Accept the correct 3SF approximation 605000. Following use of 806000 from part (a), accept 604400 (or its correct 3SF form 604000) when the correct forbidden count 201600 is subtracted.
Komentarz. Paper 2: calculator and AA HL formula booklet allowed. Unless stated otherwise give exact answers or three significant figures, retaining unrounded intermediate values.