Worked solution · Exam P

SOA Exam P Sample Question #524 Solution

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.

Independent analysis by ActuaryProof · reviewed July 31, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

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Independent solution

How to solve this Combinatorial Probability question

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.

n_{H\cap C}=9+9-15=3

n_{H\setminus C}=n_{C\setminus H}=6,\qquad n_{(H\cup C)^c}=35-15=20

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=\binom{6}{4}\binom{6}{1}\binom{3}{2}\binom{20}{1}

Compute

Evaluate each independent selection factor.

\binom{6}{4}=15,\quad \binom{6}{1}=6,\quad \binom{3}{2}=3,\quad \binom{20}{1}=20

N=15(6)(3)(20)=5{,}400

Answer

There are 5,400 eligible groups.

\boxed{5{,}400\quad\text{(C)}}

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Error diagnosis

Why the other letters tempt

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

Answer and calculation

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

The three candidate pools are disjoint, and only the analyst group carries an additional role assignment.

k_P=2,\qquad k_M=2,\qquad k_A=3

Model

Choose each role group without order, then choose the recorder from the three analysts already selected.

N=\binom{5}{2}\binom{6}{2}\binom{8}{3}(3)

Compute

Evaluate the combination factors and the recorder choice.

\binom{5}{2}=10,\quad \binom{6}{2}=15,\quad \binom{8}{3}=56

N=10(15)(56)(3)=25{,}200

Answer

There are 25,200 distinct team-and-recorder outcomes.

\boxed{25{,}200\quad\text{(B)}}

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How to solve this Combinatorial Probability question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: expedition team with a designated recorder

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability