This Exam P sample reference tests Combinatorial Probability. Inclusion-exclusion leaves 6 people in each one-condition class, 3 in the overlap, and 20 outside the union. The required product of combination counts is 5,400, so choice C is correct.
How to solve this Combinatorial Probability question
Setup
Setup
Partition the population into four disjoint cells. Inclusion-exclusion determines the shared cell first, after which the two single-condition cells and the outside cell follow.
nH∩C=9+9−15=3
nH∖C=nC∖H=6,n(H∪C)c=35−15=20
Model
Model
Seven specified selections come from the three condition cells, so the eighth member must come from outside their union. The cells are disjoint, allowing their combination counts to be multiplied.
N=(46)(16)(23)(120)
Compute
Compute
Evaluate each independent selection factor.
(46)=15,(16)=6,(23)=3,(120)=20
N=15(6)(3)(20)=5,400
Answer
Answer
There are 5,400 eligible groups.
5,400(C)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis is only C(6,4)C(6,1)=90; it stops after selecting from the two single-condition cells and leaves the group incomplete.
BThis adds the overlap selection but omits the required eighth member from the 20-person outside cell: 15(6)(3)=270.
DThis treats both marginal counts of 9 as exclusive cells and then also retains the 3-person overlap. Its inconsistent partition gives C(9,4)C(9,1)C(3,2)(14)=47,628.
EThis multiplies C(9,4)²C(15,1)=238,140, selecting from both overlapping marginal sets and then again from their union instead of using four disjoint cells.
Original practice · fully worked
Original variant: expedition team with a designated recorder
An expedition draws from three disjoint candidate pools: 5 route planners, 6 mechanics, and 8 field analysts. A team contains 2 planners, 2 mechanics, and 3 analysts. Exactly one of the selected analysts is designated as recorder. How many distinct teams with recorder assignments are possible?
A 8,400
B 25,200
C 50,400
D 67,200
E 100,800
Variant answer in brief
Choose the three role groups and then designate one of the three selected analysts. The count is C(5,2)C(6,2)C(8,3)(3)=25,200, so choice B is correct.
Setup
Setup
The three candidate pools are disjoint, and only the analyst group carries an additional role assignment.
kP=2,kM=2,kA=3
Model
Model
Choose each role group without order, then choose the recorder from the three analysts already selected.
N=(25)(26)(38)(3)
Compute
Compute
Evaluate the combination factors and the recorder choice.
(25)=10,(26)=15,(38)=56
N=10(15)(56)(3)=25,200
Answer
Answer
There are 25,200 distinct team-and-recorder outcomes.
The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.
