If the set U has 10 elements, how many members of P(U) are disjoint from a fixed 4-element subset of U?
Answer and explanation
Correct answer: 64
Let the fixed 4-element subset be F. A subset of U is disjoint from F precisely when it contains none of the elements of F. Therefore, its elements can be selected only from the remaining 10 − 4 = 6 elements. Each of these six elements has two independent choices: included or excluded. Hence the number of such subsets is 2^6 = 64. This is a direct application of the rule that an n-element set has 2^n subsets, with the forbidden elements removed.
Frequently asked questions
What is the correct answer to this question?
64
Why is this the correct answer?
Let the fixed 4-element subset be F. A subset of U is disjoint from F precisely when it contains none of the elements of F. Therefore, its elements can be selected only from the remaining 10 − 4 = 6 elements. Each of these six elements has two independent choices: included or excluded. Hence the number of such subsets is 2^6 = 64. This is a direct application of the rule that an n-element set has 2^n subsets, with the forbidden elements removed.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
