Let awards in event A is 48 and awards in event B is 25 and awards in event C is 18 and also n(A ∪ B ∪ C) = 60, n(A ⋂ B ⋂ C) = 5, then how many got exactly two awards is?
21
25
24
23
The Correct Option is A
Solution and Explanation
Let the total number of men be represented by the set \( A \cup B \cup C = 60 \) where: - \( |A| = 48 \) (men who received medals in event A) - \( |B| = 25 \) (men who received medals in event B) - \( |C| = 18 \) (men who received medals in event C) - \( |A \cup B \cup C| = 60 \) (total number of men) The number of men who received medals in all three events is given by: \[ |A \cap B \cap C| = 5 \]
We need to find how many men received medals in exactly two events, which is calculated by: \[ |A \cap B| + |B \cap C| + |C \cap A| - 3|A \cap B \cap C| \] Using the inclusion-exclusion principle, we get: \[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C| \] Substituting the values we know: \[ 60 = 48 + 25 + 18 - |A \cap B| - |B \cap C| - |C \cap A| + 5 \] \[ 60 = 91 - |A \cap B| - |B \cap C| - |C \cap A| + 5 \] \[ |A \cap B| + |B \cap C| + |C \cap A| = 36 \] Now, to find the number of men who received exactly two medals, we use the formula: \[ \text{No. of men who received exactly 2 medals} = |A \cap B| + |B \cap C| + |C \cap A| - 3|A \cap B \cap C| \] Substituting the values: \[ \text{No. of men who received exactly 2 medals} = 36 - 15 = 21 \] Thus, the number of men who received exactly two medals is 21.
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Concepts Used:
Sets
In mathematics, a set is a well-defined collection of objects. Sets are named and demonstrated using capital letter. In the set theory, the elements that a set comprises can be any sort of thing: people, numbers, letters of the alphabet, shapes, variables, etc.
Read More: Set Theory
Elements of a Set:
The items existing in a set are commonly known to be either elements or members of a set. The elements of a set are bounded in curly brackets separated by commas.
Read Also: Set Operation
Cardinal Number of a Set:
The cardinal number, cardinality, or order of a set indicates the total number of elements in the set.
Read More: Types of Sets