Worked solution · Exam P
This Exam P sample reference tests Counting Methods. The favorable assignments are the derangements of four labeled objects. Inclusion-exclusion gives nine derangements among 24 permutations, so the probability is 9/24=0.375 and choice D.
Independent solution
Setup
There are four factorial equally likely assignments. A favorable assignment has no fixed point.
Model
Apply inclusion-exclusion to the events that each labeled object returns to its matching position.
Compute
Evaluate the derangement count and divide by all assignments.
Answer
The no-match probability is 0.375.
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Original practice · fully worked
Six distinct diagnostic steps are arranged uniformly at random in a line. Step A may not be adjacent to step B, and step C may not be adjacent to step D. Calculate the probability that neither forbidden adjacency occurs.
Each specified adjacent pair forms a two-order block in 2(5!) arrangements, and both blocks occur in 4(4!) arrangements. Inclusion-exclusion leaves 336 of 720 orders, giving 7/15 and choice C.
Setup
Count all linear orders of the six labeled steps.
Model
Treat each forbidden adjacency as a block. Either single block has two internal orders, while two simultaneous blocks have four.
Compute
Use inclusion-exclusion to remove orders containing either forbidden block.
Answer
Neither forbidden pair is adjacent with probability 7/15.
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