3 questions from 2024
PARAGRAPH – 1
Let \( S = \{1, 2, 3, 4, 5, 6\} \) and \( X \) be the set of all relations \( R \) from \( S \) to \( S \) that satisfy both the following properties:
i. \( R \) has exactly \( 6 \) elements.
ii. For each \( (a, b) \in R \), we have \( |a-b| \ge 2 \).
Let \( Y = \{\, R \in X : \text{ The range of } R \text{ has exactly one element} \,\} \) and \( Z = \{\, R \in X : R \text{ is a function from } S \text{ to } S \,\} \).
Let \( n(A) \) denote the number of elements in a set \( A \).
If $n(X) = {}^m C_6$, then the value of $m$ is ______________.
PARAGRAPH – 1
Let \( S = \{1, 2, 3, 4, 5, 6\} \) and \( X \) be the set of all relations \( R \) from \( S \) to \( S \) that satisfy both the following properties:
i. \( R \) has exactly \( 6 \) elements.
ii. For each \( (a, b) \in R \), we have \( |a-b| \ge 2 \).
Let \( Y = \{\, R \in X : \text{ The range of } R \text{ has exactly one element} \,\} \) and \( Z = \{\, R \in X : R \text{ is a function from } S \text{ to } S \,\} \).
Let \( n(A) \) denote the number of elements in a set \( A \).
If the value of \( n(Y) + n(Z) \) is \( k^{2} \), then \( |k| \) is __________.
A group of 9 students, $s_1, s_2, \ldots, s_9$, is to be divided to form three teams $X, Y$, and $Z$ of sizes 2, 3, and 4, respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is ______________.
