In a regional administrative office of 80 officers, 45 officers completed a training cours… — State PSC Exam

In a regional administrative office of $80$ officers, $45$ officers completed a training course in Data Analytics, $40$ officers completed a course in Public Policy, and $15$ officers completed both training courses. How many officers completed neither of the two training courses?

$10$ Answer
B
$25$
C
$15$
D
$20$

Answer

10 officers completed neither of the two training courses.

To find the number of officers who completed neither course, calculate the union of the two sets using inclusion-exclusion:

n(\text{Data Analytics} \cup \text{Public Policy}) = 45 + 40 - 15 = 70

. The number of officers who completed neither course is the complement of this union relative to the total group:

80 - 70 = 10

Step-by-Step Solution

Identify the given set values and total universe

Total officers

n(U) = 80

, Data Analytics

n(A) = 45

, Public Policy

n(B) = 40

, Both

n(A \cap B) = 15

Establish baseline data to apply the set inclusion-exclusion principle.

Calculate the number of officers who completed at least one training course using the principle of inclusion-exclusion

n(A \cup B) = n(A) + n(B) - n(A \cap B) = 45 + 40 - 15 = 70

Avoid double-counting the officers who completed both courses.

Calculate the number of officers who completed neither course

n(\text{Neither}) = n(U) - n(A \cup B) = 80 - 70 = 10

Subtract the union of the two sets from the total population to find the complement set.

Key Concept

Two-set Principle of Inclusion-Exclusion and Complementary Sets

Estimated Time:45s

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