Worked solution · Exam P
This Exam P sample reference tests Combinatorics. Choose three of five columns, then assign three distinct rows from six in order. The count is C(5,3)(6)(5)(4)=1200, choice C.
Independent solution
Setup
Choose which three of the five columns are occupied, then assign a distinct row to each chosen column.
Model
After the column set is fixed, there are 6×5×4 injective row assignments.
Compute
Multiplying 10 column choices by 120 row assignments gives 1200 valid placements.
Answer
The number of arrangements is 1200, corresponding to choice C.
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Original practice · fully worked
Eight analysts sit around a round table, with rotations regarded as identical. Two designated analysts must not sit next to each other. How many valid seating arrangements are possible?
There are 7! circular arrangements in total. Treating the designated pair as a two-person block gives 2(6!) adjacent arrangements, leaving 3600.
Setup
Count all circular seatings of eight analysts, then subtract those in which the designated pair is adjacent.
Model
There are 7! total circular arrangements. Treating the designated pair as a block gives 2×6! adjacent arrangements.
Compute
Subtracting 1440 forbidden arrangements from 5040 total arrangements leaves 3600.
Answer
The number of valid circular seatings is 3600, corresponding to choice C.
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