Q192 / 270 · NDA 2025 (Phase 2) · Mathematics
Set $X$ contains $3n$ elements and set $Y$ contains $2n$ elements, and they have $n$ elements in common. How many elements does $(X - Y) \times (Y - X)$ have?
- A$5n^2$
- B$4n^2$
- C$3n^2$
- Correct answerD$2n^2$
Solution
Given $n(X) = 3n$, $n(Y) = 2n$, and $n(X \cap Y) = n$.
The number of elements in $X - Y$ is:
$n(X - Y) = n(X) - n(X \cap Y) = 3n - n = 2n$
The number of elements in $Y - X$ is:
$n(Y - X) = n(Y) - n(X \cap Y) = 2n - n = n$
The number of elements in the Cartesian product $(X - Y) \times (Y - X)$ is:
$n((X - Y) \times (Y - X)) = n(X - Y) \times n(Y - X)$
Substituting the values:
$n((X - Y) \times (Y - X)) = 2n \times n = 2n^2$
Answer: $2n^2$
The number of elements in $X - Y$ is:
$n(X - Y) = n(X) - n(X \cap Y) = 3n - n = 2n$
The number of elements in $Y - X$ is:
$n(Y - X) = n(Y) - n(X \cap Y) = 2n - n = n$
The number of elements in the Cartesian product $(X - Y) \times (Y - X)$ is:
$n((X - Y) \times (Y - X)) = n(X - Y) \times n(Y - X)$
Substituting the values:
$n((X - Y) \times (Y - X)) = 2n \times n = 2n^2$
Answer: $2n^2$
Hence, the correct answer is option: 'D'.