If \(A=\{1,2,3,4\}\), how many proper subsets of \(A\) have exactly three elements?
Answer and explanation
Correct answer: 4
A three-element subset is formed by choosing 3 of the 4 elements of \(A\). The number of such choices is \(\binom{4}{3}=\frac{4!}{3!1!}=4\). Each resulting set has only three elements, whereas \(A\) has four, so none of them equals \(A\); consequently, all four are proper subsets. They are \{1,2,3\}, \{1,2,4\}, \{1,3,4\}, and \{2,3,4\}.
Frequently asked questions
What is the correct answer to this question?
4
Why is this the correct answer?
A three-element subset is formed by choosing 3 of the 4 elements of \(A\). The number of such choices is \(\binom{4}{3}=\frac{4!}{3!1!}=4\). Each resulting set has only three elements, whereas \(A\) has four, so none of them equals \(A\); consequently, all four are proper subsets. They are \{1,2,3\}, \{1,2,4\}, \{1,3,4\}, and \{2,3,4\}.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
